Punctured Limit Witness and Point-Value Depth
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261007-advance-20261009-r03-function-limit-foundations · 2026-10-07 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Exact Punctured Counterexample to a Failed δ
On a punctured real domain, f(x)=L+(1)(x−A) for x≠A. Given ε=1/1>0 and failed δ=2/1>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
(1/1, 1/1)
The witness pair is (1/1, 1/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-invalid-delta-witness · candidate
2. Exact Punctured Counterexample to a Failed δ
On a punctured real domain, f(x)=L+(-2)(x−A) for x≠A. Given ε=2/2>0 and failed δ=6/4>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
(1/2, 1/1)
The witness pair is (1/2, 1/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-invalid-delta-witness · candidate
3. Exact Punctured Counterexample to a Failed δ
On a punctured real domain, f(x)=L+(3)(x−A) for x≠A. Given ε=3/3>0 and failed δ=12/9>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
(1/3, 1/1)
The witness pair is (1/3, 1/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-invalid-delta-witness · candidate
4. Exact Punctured Counterexample to a Failed δ
On a punctured real domain, f(x)=L+(-4)(x−A) for x≠A. Given ε=4/4>0 and failed δ=8/16>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
(1/4, 1/1)
The witness pair is (1/4, 1/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-invalid-delta-witness · candidate
5. Exact Punctured Counterexample to a Failed δ
On a punctured real domain, f(x)=L+(5)(x−A) for x≠A. Given ε=5/5>0 and failed δ=15/25>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
(1/5, 1/1)
The witness pair is (1/5, 1/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-invalid-delta-witness · candidate
6. Exact Punctured Counterexample to a Failed δ
On a punctured real domain, f(x)=L+(-6)(x−A) for x≠A. Given ε=6/1>0 and failed δ=24/6>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
(1/1, 6/1)
The witness pair is (1/1, 6/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-invalid-delta-witness · candidate
7. Exact Punctured Counterexample to a Failed δ
On a punctured real domain, f(x)=L+(7)(x−A) for x≠A. Given ε=7/2>0 and failed δ=14/14>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
(1/2, 7/2)
The witness pair is (1/2, 7/2): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-invalid-delta-witness · candidate
8. Exact Punctured Counterexample to a Failed δ
On a punctured real domain, f(x)=L+(-8)(x−A) for x≠A. Given ε=8/3>0 and failed δ=24/24>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
(1/3, 8/3)
The witness pair is (1/3, 8/3): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-invalid-delta-witness · candidate
9. Exact Punctured Counterexample to a Failed δ
On a punctured real domain, f(x)=L+(9)(x−A) for x≠A. Given ε=9/4>0 and failed δ=36/36>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
(1/4, 9/4)
The witness pair is (1/4, 9/4): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-invalid-delta-witness · candidate
10. Exact Punctured Counterexample to a Failed δ
On a punctured real domain, f(x)=L+(-10)(x−A) for x≠A. Given ε=10/5>0 and failed δ=20/50>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
(1/5, 2/1)
The witness pair is (1/5, 2/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-invalid-delta-witness · candidate
11. Exact Punctured Counterexample to a Failed δ
On a punctured real domain, f(x)=L+(11)(x−A) for x≠A. Given ε=11/1>0 and failed δ=33/11>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
(1/1, 11/1)
The witness pair is (1/1, 11/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-invalid-delta-witness · candidate
12. Exact Punctured Counterexample to a Failed δ
On a punctured real domain, f(x)=L+(-12)(x−A) for x≠A. Given ε=12/2>0 and failed δ=48/24>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
(1/2, 6/1)
The witness pair is (1/2, 6/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-invalid-delta-witness · candidate
13. Exact Punctured Counterexample to a Failed δ
On a punctured real domain, f(x)=L+(13)(x−A) for x≠A. Given ε=13/3>0 and failed δ=26/39>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
(1/3, 13/3)
The witness pair is (1/3, 13/3): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-invalid-delta-witness · candidate
14. Exact Punctured Counterexample to a Failed δ
On a punctured real domain, f(x)=L+(-14)(x−A) for x≠A. Given ε=14/4>0 and failed δ=42/56>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
(1/4, 7/2)
The witness pair is (1/4, 7/2): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-invalid-delta-witness · candidate
15. Exact Punctured Counterexample to a Failed δ
On a punctured real domain, f(x)=L+(15)(x−A) for x≠A. Given ε=15/5>0 and failed δ=60/75>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
(1/5, 3/1)
The witness pair is (1/5, 3/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-invalid-delta-witness · candidate
16. Exact Punctured Counterexample to a Failed δ
On a punctured real domain, f(x)=L+(-16)(x−A) for x≠A. Given ε=16/1>0 and failed δ=32/16>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
(1/1, 16/1)
The witness pair is (1/1, 16/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-invalid-delta-witness · candidate
17. Exact Punctured Counterexample to a Failed δ
On a punctured real domain, f(x)=L+(17)(x−A) for x≠A. Given ε=17/2>0 and failed δ=51/34>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
(1/2, 17/2)
The witness pair is (1/2, 17/2): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-invalid-delta-witness · candidate
18. Exact Punctured Counterexample to a Failed δ
On a punctured real domain, f(x)=L+(-18)(x−A) for x≠A. Given ε=18/3>0 and failed δ=72/54>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
(1/3, 6/1)
The witness pair is (1/3, 6/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-invalid-delta-witness · candidate
19. Exact Punctured Counterexample to a Failed δ
On a punctured real domain, f(x)=L+(19)(x−A) for x≠A. Given ε=19/4>0 and failed δ=38/76>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
(1/4, 19/4)
The witness pair is (1/4, 19/4): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-invalid-delta-witness · candidate
20. Exact Punctured Counterexample to a Failed δ
On a punctured real domain, f(x)=L+(-20)(x−A) for x≠A. Given ε=20/5>0 and failed δ=60/100>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
(1/5, 4/1)
The witness pair is (1/5, 4/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-invalid-delta-witness · candidate
21. Exact Punctured Counterexample to a Failed δ
On a punctured real domain, f(x)=L+(21)(x−A) for x≠A. Given ε=21/1>0 and failed δ=84/21>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
(1/1, 21/1)
The witness pair is (1/1, 21/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-invalid-delta-witness · candidate
22. Exact Punctured Counterexample to a Failed δ
On a punctured real domain, f(x)=L+(-22)(x−A) for x≠A. Given ε=22/2>0 and failed δ=44/44>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
(1/2, 11/1)
The witness pair is (1/2, 11/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-invalid-delta-witness · candidate
23. Exact Punctured Counterexample to a Failed δ
On a punctured real domain, f(x)=L+(23)(x−A) for x≠A. Given ε=23/3>0 and failed δ=69/69>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
(1/3, 23/3)
The witness pair is (1/3, 23/3): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-invalid-delta-witness · candidate
24. Exact Punctured Counterexample to a Failed δ
On a punctured real domain, f(x)=L+(-24)(x−A) for x≠A. Given ε=24/4>0 and failed δ=96/96>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
(1/4, 6/1)
The witness pair is (1/4, 6/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-invalid-delta-witness · candidate
25. Exact Punctured Counterexample to a Failed δ
On a punctured real domain, f(x)=L+(25)(x−A) for x≠A. Given ε=25/5>0 and failed δ=50/125>0 with |m|δ>ε, give (h,error) for the specified positive witness h=ε/|m|. It must satisfy 0<h<δ but fail |f(A+h)−L|<ε. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
(1/5, 5/1)
The witness pair is (1/5, 5/1): h lies in the strict punctured interior, but the error equals ε rather than being smaller. The target is L, not an undefined f(A), and the nonzero-slope premise is required.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-invalid-delta-witness · candidate
26. Ordered Center-Value and Punctured-Limit Pair
The full-real function has affine expression (-12)x+(-36) away from a=-2, and separately defined center value -12. Give (center value,punctured two-sided limit).
View answer and explanation
(-12, -12)
The explicit point value and affine approach value give (-12, -12), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-center-versus-limit · candidate
27. Ordered Center-Value and Punctured-Limit Pair
The full-real function has affine expression (-11)x+(-22) away from a=-1, and separately defined center value -9. Give (center value,punctured two-sided limit).
View answer and explanation
(-9, -11)
The explicit point value and affine approach value give (-9, -11), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-center-versus-limit · candidate
28. Ordered Center-Value and Punctured-Limit Pair
The full-real function has affine expression (-10)x+(-10) away from a=0, and separately defined center value -7. Give (center value,punctured two-sided limit).
View answer and explanation
(-7, -10)
The explicit point value and affine approach value give (-7, -10), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-center-versus-limit · candidate
29. Ordered Center-Value and Punctured-Limit Pair
The full-real function has affine expression (-9)x+(0) away from a=1, and separately defined center value -9. Give (center value,punctured two-sided limit).
View answer and explanation
(-9, -9)
The explicit point value and affine approach value give (-9, -9), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-center-versus-limit · candidate
30. Ordered Center-Value and Punctured-Limit Pair
The full-real function has affine expression (-8)x+(8) away from a=2, and separately defined center value -3. Give (center value,punctured two-sided limit).
View answer and explanation
(-3, -8)
The explicit point value and affine approach value give (-3, -8), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-center-versus-limit · candidate
31. Ordered Center-Value and Punctured-Limit Pair
The full-real function has affine expression (-7)x+(-21) away from a=-2, and separately defined center value -6. Give (center value,punctured two-sided limit).
View answer and explanation
(-6, -7)
The explicit point value and affine approach value give (-6, -7), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-center-versus-limit · candidate
32. Ordered Center-Value and Punctured-Limit Pair
The full-real function has affine expression (-6)x+(-12) away from a=-1, and separately defined center value -6. Give (center value,punctured two-sided limit).
View answer and explanation
(-6, -6)
The explicit point value and affine approach value give (-6, -6), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-center-versus-limit · candidate
33. Ordered Center-Value and Punctured-Limit Pair
The full-real function has affine expression (-5)x+(-5) away from a=0, and separately defined center value -2. Give (center value,punctured two-sided limit).
View answer and explanation
(-2, -5)
The explicit point value and affine approach value give (-2, -5), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-center-versus-limit · candidate
34. Ordered Center-Value and Punctured-Limit Pair
The full-real function has affine expression (-4)x+(0) away from a=1, and separately defined center value 0. Give (center value,punctured two-sided limit).
View answer and explanation
(0, -4)
The explicit point value and affine approach value give (0, -4), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-center-versus-limit · candidate
35. Ordered Center-Value and Punctured-Limit Pair
The full-real function has affine expression (-3)x+(3) away from a=2, and separately defined center value -3. Give (center value,punctured two-sided limit).
View answer and explanation
(-3, -3)
The explicit point value and affine approach value give (-3, -3), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-center-versus-limit · candidate
36. Ordered Center-Value and Punctured-Limit Pair
The full-real function has affine expression (-2)x+(-6) away from a=-2, and separately defined center value -1. Give (center value,punctured two-sided limit).
View answer and explanation
(-1, -2)
The explicit point value and affine approach value give (-1, -2), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-center-versus-limit · candidate
37. Ordered Center-Value and Punctured-Limit Pair
The full-real function has affine expression (-1)x+(-2) away from a=-1, and separately defined center value 1. Give (center value,punctured two-sided limit).
View answer and explanation
(1, -1)
The explicit point value and affine approach value give (1, -1), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-center-versus-limit · candidate
38. Ordered Center-Value and Punctured-Limit Pair
The full-real function has affine expression (0)x+(0) away from a=0, and separately defined center value 0. Give (center value,punctured two-sided limit).
View answer and explanation
(0, 0)
The explicit point value and affine approach value give (0, 0), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-center-versus-limit · candidate
39. Ordered Center-Value and Punctured-Limit Pair
The full-real function has affine expression (1)x+(0) away from a=1, and separately defined center value 5. Give (center value,punctured two-sided limit).
View answer and explanation
(5, 1)
The explicit point value and affine approach value give (5, 1), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-center-versus-limit · candidate
40. Ordered Center-Value and Punctured-Limit Pair
The full-real function has affine expression (2)x+(-2) away from a=2, and separately defined center value 7. Give (center value,punctured two-sided limit).
View answer and explanation
(7, 2)
The explicit point value and affine approach value give (7, 2), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-center-versus-limit · candidate
41. Ordered Center-Value and Punctured-Limit Pair
The full-real function has affine expression (3)x+(9) away from a=-2, and separately defined center value 3. Give (center value,punctured two-sided limit).
View answer and explanation
(3, 3)
The explicit point value and affine approach value give (3, 3), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-center-versus-limit · candidate
42. Ordered Center-Value and Punctured-Limit Pair
The full-real function has affine expression (4)x+(8) away from a=-1, and separately defined center value 6. Give (center value,punctured two-sided limit).
View answer and explanation
(6, 4)
The explicit point value and affine approach value give (6, 4), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-center-versus-limit · candidate
43. Ordered Center-Value and Punctured-Limit Pair
The full-real function has affine expression (5)x+(5) away from a=0, and separately defined center value 8. Give (center value,punctured two-sided limit).
View answer and explanation
(8, 5)
The explicit point value and affine approach value give (8, 5), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-center-versus-limit · candidate
44. Ordered Center-Value and Punctured-Limit Pair
The full-real function has affine expression (6)x+(0) away from a=1, and separately defined center value 6. Give (center value,punctured two-sided limit).
View answer and explanation
(6, 6)
The explicit point value and affine approach value give (6, 6), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-center-versus-limit · candidate
45. Ordered Center-Value and Punctured-Limit Pair
The full-real function has affine expression (7)x+(-7) away from a=2, and separately defined center value 12. Give (center value,punctured two-sided limit).
View answer and explanation
(12, 7)
The explicit point value and affine approach value give (12, 7), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-center-versus-limit · candidate
46. Ordered Center-Value and Punctured-Limit Pair
The full-real function has affine expression (8)x+(24) away from a=-2, and separately defined center value 9. Give (center value,punctured two-sided limit).
View answer and explanation
(9, 8)
The explicit point value and affine approach value give (9, 8), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-center-versus-limit · candidate
47. Ordered Center-Value and Punctured-Limit Pair
The full-real function has affine expression (9)x+(18) away from a=-1, and separately defined center value 9. Give (center value,punctured two-sided limit).
View answer and explanation
(9, 9)
The explicit point value and affine approach value give (9, 9), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-center-versus-limit · candidate
48. Ordered Center-Value and Punctured-Limit Pair
The full-real function has affine expression (10)x+(10) away from a=0, and separately defined center value 13. Give (center value,punctured two-sided limit).
View answer and explanation
(13, 10)
The explicit point value and affine approach value give (13, 10), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-center-versus-limit · candidate
49. Ordered Center-Value and Punctured-Limit Pair
The full-real function has affine expression (11)x+(0) away from a=1, and separately defined center value 15. Give (center value,punctured two-sided limit).
View answer and explanation
(15, 11)
The explicit point value and affine approach value give (15, 11), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-center-versus-limit · candidate
50. Ordered Center-Value and Punctured-Limit Pair
The full-real function has affine expression (12)x+(-12) away from a=2, and separately defined center value 12. Give (center value,punctured two-sided limit).
View answer and explanation
(12, 12)
The explicit point value and affine approach value give (12, 12), preserving order. Changing the center does not alter the punctured limit; unequal values prevent center continuity.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-center-versus-limit · candidate
51. Ordered Left and Right Punctured Limits
On R∖{-2}, the branch for x<a is (1)x+(-10), and for x>a is (-1)x+(-14). Give (left limit,right limit), keeping the order.
View answer and explanation
(-12, -12)
Evaluate the two affine approach values to obtain (-12, -12). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-one-sided-pair · candidate
52. Ordered Left and Right Punctured Limits
On R∖{-1}, the branch for x<a is (2)x+(-9), and for x>a is (-2)x+(10). Give (left limit,right limit), keeping the order.
View answer and explanation
(-11, 12)
Evaluate the two affine approach values to obtain (-11, 12). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-one-sided-pair · candidate
53. Ordered Left and Right Punctured Limits
On R∖{0}, the branch for x<a is (3)x+(-10), and for x>a is (-1)x+(11). Give (left limit,right limit), keeping the order.
View answer and explanation
(-10, 11)
Evaluate the two affine approach values to obtain (-10, 11). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-one-sided-pair · candidate
54. Ordered Left and Right Punctured Limits
On R∖{1}, the branch for x<a is (1)x+(-10), and for x>a is (-2)x+(-7). Give (left limit,right limit), keeping the order.
View answer and explanation
(-9, -9)
Evaluate the two affine approach values to obtain (-9, -9). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-one-sided-pair · candidate
55. Ordered Left and Right Punctured Limits
On R∖{-2}, the branch for x<a is (2)x+(-4), and for x>a is (-1)x+(7). Give (left limit,right limit), keeping the order.
View answer and explanation
(-8, 9)
Evaluate the two affine approach values to obtain (-8, 9). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-one-sided-pair · candidate
56. Ordered Left and Right Punctured Limits
On R∖{-1}, the branch for x<a is (3)x+(-4), and for x>a is (-2)x+(6). Give (left limit,right limit), keeping the order.
View answer and explanation
(-7, 8)
Evaluate the two affine approach values to obtain (-7, 8). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-one-sided-pair · candidate
57. Ordered Left and Right Punctured Limits
On R∖{0}, the branch for x<a is (1)x+(-6), and for x>a is (-1)x+(-6). Give (left limit,right limit), keeping the order.
View answer and explanation
(-6, -6)
Evaluate the two affine approach values to obtain (-6, -6). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-one-sided-pair · candidate
58. Ordered Left and Right Punctured Limits
On R∖{1}, the branch for x<a is (2)x+(-7), and for x>a is (-2)x+(8). Give (left limit,right limit), keeping the order.
View answer and explanation
(-5, 6)
Evaluate the two affine approach values to obtain (-5, 6). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-one-sided-pair · candidate
59. Ordered Left and Right Punctured Limits
On R∖{-2}, the branch for x<a is (3)x+(2), and for x>a is (-1)x+(3). Give (left limit,right limit), keeping the order.
View answer and explanation
(-4, 5)
Evaluate the two affine approach values to obtain (-4, 5). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-one-sided-pair · candidate
60. Ordered Left and Right Punctured Limits
On R∖{-1}, the branch for x<a is (1)x+(-2), and for x>a is (-2)x+(-5). Give (left limit,right limit), keeping the order.
View answer and explanation
(-3, -3)
Evaluate the two affine approach values to obtain (-3, -3). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-one-sided-pair · candidate
61. Ordered Left and Right Punctured Limits
On R∖{0}, the branch for x<a is (2)x+(-2), and for x>a is (-1)x+(3). Give (left limit,right limit), keeping the order.
View answer and explanation
(-2, 3)
Evaluate the two affine approach values to obtain (-2, 3). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-one-sided-pair · candidate
62. Ordered Left and Right Punctured Limits
On R∖{1}, the branch for x<a is (3)x+(-4), and for x>a is (-2)x+(4). Give (left limit,right limit), keeping the order.
View answer and explanation
(-1, 2)
Evaluate the two affine approach values to obtain (-1, 2). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-one-sided-pair · candidate
63. Ordered Left and Right Punctured Limits
On R∖{-2}, the branch for x<a is (1)x+(2), and for x>a is (-1)x+(-2). Give (left limit,right limit), keeping the order.
View answer and explanation
(0, 0)
Evaluate the two affine approach values to obtain (0, 0). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-one-sided-pair · candidate
64. Ordered Left and Right Punctured Limits
On R∖{-1}, the branch for x<a is (2)x+(3), and for x>a is (-2)x+(-2). Give (left limit,right limit), keeping the order.
View answer and explanation
(1, 0)
Evaluate the two affine approach values to obtain (1, 0). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-one-sided-pair · candidate
65. Ordered Left and Right Punctured Limits
On R∖{0}, the branch for x<a is (3)x+(2), and for x>a is (-1)x+(-1). Give (left limit,right limit), keeping the order.
View answer and explanation
(2, -1)
Evaluate the two affine approach values to obtain (2, -1). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-one-sided-pair · candidate
66. Ordered Left and Right Punctured Limits
On R∖{1}, the branch for x<a is (1)x+(2), and for x>a is (-2)x+(5). Give (left limit,right limit), keeping the order.
View answer and explanation
(3, 3)
Evaluate the two affine approach values to obtain (3, 3). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-one-sided-pair · candidate
67. Ordered Left and Right Punctured Limits
On R∖{-2}, the branch for x<a is (2)x+(8), and for x>a is (-1)x+(-5). Give (left limit,right limit), keeping the order.
View answer and explanation
(4, -3)
Evaluate the two affine approach values to obtain (4, -3). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-one-sided-pair · candidate
68. Ordered Left and Right Punctured Limits
On R∖{-1}, the branch for x<a is (3)x+(8), and for x>a is (-2)x+(-6). Give (left limit,right limit), keeping the order.
View answer and explanation
(5, -4)
Evaluate the two affine approach values to obtain (5, -4). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-one-sided-pair · candidate
69. Ordered Left and Right Punctured Limits
On R∖{0}, the branch for x<a is (1)x+(6), and for x>a is (-1)x+(6). Give (left limit,right limit), keeping the order.
View answer and explanation
(6, 6)
Evaluate the two affine approach values to obtain (6, 6). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-one-sided-pair · candidate
70. Ordered Left and Right Punctured Limits
On R∖{1}, the branch for x<a is (2)x+(5), and for x>a is (-2)x+(-4). Give (left limit,right limit), keeping the order.
View answer and explanation
(7, -6)
Evaluate the two affine approach values to obtain (7, -6). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-one-sided-pair · candidate
71. Ordered Left and Right Punctured Limits
On R∖{-2}, the branch for x<a is (3)x+(14), and for x>a is (-1)x+(-9). Give (left limit,right limit), keeping the order.
View answer and explanation
(8, -7)
Evaluate the two affine approach values to obtain (8, -7). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-one-sided-pair · candidate
72. Ordered Left and Right Punctured Limits
On R∖{-1}, the branch for x<a is (1)x+(10), and for x>a is (-2)x+(7). Give (left limit,right limit), keeping the order.
View answer and explanation
(9, 9)
Evaluate the two affine approach values to obtain (9, 9). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-one-sided-pair · candidate
73. Ordered Left and Right Punctured Limits
On R∖{0}, the branch for x<a is (2)x+(10), and for x>a is (-1)x+(-9). Give (left limit,right limit), keeping the order.
View answer and explanation
(10, -9)
Evaluate the two affine approach values to obtain (10, -9). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-one-sided-pair · candidate
74. Ordered Left and Right Punctured Limits
On R∖{1}, the branch for x<a is (3)x+(8), and for x>a is (-2)x+(-8). Give (left limit,right limit), keeping the order.
View answer and explanation
(11, -10)
Evaluate the two affine approach values to obtain (11, -10). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-one-sided-pair · candidate
75. Ordered Left and Right Punctured Limits
On R∖{-2}, the branch for x<a is (1)x+(14), and for x>a is (-1)x+(10). Give (left limit,right limit), keeping the order.
View answer and explanation
(12, 12)
Evaluate the two affine approach values to obtain (12, 12). A finite two-sided limit requires their agreement, but each one-sided value remains separately meaningful.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-one-sided-pair · candidate
76. Original Quotient Value at a Permitted Punctured Point
The original quotient (x²−(-12)²)/(x−(-12)) has punctured real domain x≠a. Evaluate it at the permitted x=-11. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-23/1
The denominator is nonzero here, so the exact original quotient value is -23/1. This does not fill the excluded center or assign a division-by-zero default.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-punctured-quotient-value · candidate
77. Original Quotient Value at a Permitted Punctured Point
The original quotient (x²−(-11)²)/(x−(-11)) has punctured real domain x≠a. Evaluate it at the permitted x=-13. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-24/1
The denominator is nonzero here, so the exact original quotient value is -24/1. This does not fill the excluded center or assign a division-by-zero default.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-punctured-quotient-value · candidate
78. Original Quotient Value at a Permitted Punctured Point
The original quotient (x²−(-10)²)/(x−(-10)) has punctured real domain x≠a. Evaluate it at the permitted x=-7. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-17/1
The denominator is nonzero here, so the exact original quotient value is -17/1. This does not fill the excluded center or assign a division-by-zero default.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-punctured-quotient-value · candidate
79. Original Quotient Value at a Permitted Punctured Point
The original quotient (x²−(-9)²)/(x−(-9)) has punctured real domain x≠a. Evaluate it at the permitted x=-13. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-22/1
The denominator is nonzero here, so the exact original quotient value is -22/1. This does not fill the excluded center or assign a division-by-zero default.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-punctured-quotient-value · candidate
80. Original Quotient Value at a Permitted Punctured Point
The original quotient (x²−(-8)²)/(x−(-8)) has punctured real domain x≠a. Evaluate it at the permitted x=-7. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-15/1
The denominator is nonzero here, so the exact original quotient value is -15/1. This does not fill the excluded center or assign a division-by-zero default.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-punctured-quotient-value · candidate
81. Original Quotient Value at a Permitted Punctured Point
The original quotient (x²−(-7)²)/(x−(-7)) has punctured real domain x≠a. Evaluate it at the permitted x=-9. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-16/1
The denominator is nonzero here, so the exact original quotient value is -16/1. This does not fill the excluded center or assign a division-by-zero default.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-punctured-quotient-value · candidate
82. Original Quotient Value at a Permitted Punctured Point
The original quotient (x²−(-6)²)/(x−(-6)) has punctured real domain x≠a. Evaluate it at the permitted x=-3. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-9/1
The denominator is nonzero here, so the exact original quotient value is -9/1. This does not fill the excluded center or assign a division-by-zero default.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-punctured-quotient-value · candidate
83. Original Quotient Value at a Permitted Punctured Point
The original quotient (x²−(-5)²)/(x−(-5)) has punctured real domain x≠a. Evaluate it at the permitted x=-9. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-14/1
The denominator is nonzero here, so the exact original quotient value is -14/1. This does not fill the excluded center or assign a division-by-zero default.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-punctured-quotient-value · candidate
84. Original Quotient Value at a Permitted Punctured Point
The original quotient (x²−(-4)²)/(x−(-4)) has punctured real domain x≠a. Evaluate it at the permitted x=-3. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-7/1
The denominator is nonzero here, so the exact original quotient value is -7/1. This does not fill the excluded center or assign a division-by-zero default.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-punctured-quotient-value · candidate
85. Original Quotient Value at a Permitted Punctured Point
The original quotient (x²−(-3)²)/(x−(-3)) has punctured real domain x≠a. Evaluate it at the permitted x=-5. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-8/1
The denominator is nonzero here, so the exact original quotient value is -8/1. This does not fill the excluded center or assign a division-by-zero default.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-punctured-quotient-value · candidate
86. Original Quotient Value at a Permitted Punctured Point
The original quotient (x²−(-2)²)/(x−(-2)) has punctured real domain x≠a. Evaluate it at the permitted x=1. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-1/1
The denominator is nonzero here, so the exact original quotient value is -1/1. This does not fill the excluded center or assign a division-by-zero default.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-punctured-quotient-value · candidate
87. Original Quotient Value at a Permitted Punctured Point
The original quotient (x²−(-1)²)/(x−(-1)) has punctured real domain x≠a. Evaluate it at the permitted x=-5. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-6/1
The denominator is nonzero here, so the exact original quotient value is -6/1. This does not fill the excluded center or assign a division-by-zero default.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-punctured-quotient-value · candidate
88. Original Quotient Value at a Permitted Punctured Point
The original quotient (x²−(0)²)/(x−(0)) has punctured real domain x≠a. Evaluate it at the permitted x=1. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
1/1
The denominator is nonzero here, so the exact original quotient value is 1/1. This does not fill the excluded center or assign a division-by-zero default.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-punctured-quotient-value · candidate
89. Original Quotient Value at a Permitted Punctured Point
The original quotient (x²−(1)²)/(x−(1)) has punctured real domain x≠a. Evaluate it at the permitted x=-1. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
0/1
The denominator is nonzero here, so the exact original quotient value is 0/1. This does not fill the excluded center or assign a division-by-zero default.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-punctured-quotient-value · candidate
90. Original Quotient Value at a Permitted Punctured Point
The original quotient (x²−(2)²)/(x−(2)) has punctured real domain x≠a. Evaluate it at the permitted x=5. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
7/1
The denominator is nonzero here, so the exact original quotient value is 7/1. This does not fill the excluded center or assign a division-by-zero default.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-punctured-quotient-value · candidate
91. Original Quotient Value at a Permitted Punctured Point
The original quotient (x²−(3)²)/(x−(3)) has punctured real domain x≠a. Evaluate it at the permitted x=-1. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
2/1
The denominator is nonzero here, so the exact original quotient value is 2/1. This does not fill the excluded center or assign a division-by-zero default.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-punctured-quotient-value · candidate
92. Original Quotient Value at a Permitted Punctured Point
The original quotient (x²−(4)²)/(x−(4)) has punctured real domain x≠a. Evaluate it at the permitted x=5. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
9/1
The denominator is nonzero here, so the exact original quotient value is 9/1. This does not fill the excluded center or assign a division-by-zero default.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-punctured-quotient-value · candidate
93. Original Quotient Value at a Permitted Punctured Point
The original quotient (x²−(5)²)/(x−(5)) has punctured real domain x≠a. Evaluate it at the permitted x=3. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
8/1
The denominator is nonzero here, so the exact original quotient value is 8/1. This does not fill the excluded center or assign a division-by-zero default.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-punctured-quotient-value · candidate
94. Original Quotient Value at a Permitted Punctured Point
The original quotient (x²−(6)²)/(x−(6)) has punctured real domain x≠a. Evaluate it at the permitted x=9. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
15/1
The denominator is nonzero here, so the exact original quotient value is 15/1. This does not fill the excluded center or assign a division-by-zero default.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-punctured-quotient-value · candidate
95. Original Quotient Value at a Permitted Punctured Point
The original quotient (x²−(7)²)/(x−(7)) has punctured real domain x≠a. Evaluate it at the permitted x=3. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
10/1
The denominator is nonzero here, so the exact original quotient value is 10/1. This does not fill the excluded center or assign a division-by-zero default.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-punctured-quotient-value · candidate
96. Original Quotient Value at a Permitted Punctured Point
The original quotient (x²−(8)²)/(x−(8)) has punctured real domain x≠a. Evaluate it at the permitted x=9. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
17/1
The denominator is nonzero here, so the exact original quotient value is 17/1. This does not fill the excluded center or assign a division-by-zero default.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-punctured-quotient-value · candidate
97. Original Quotient Value at a Permitted Punctured Point
The original quotient (x²−(9)²)/(x−(9)) has punctured real domain x≠a. Evaluate it at the permitted x=7. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
16/1
The denominator is nonzero here, so the exact original quotient value is 16/1. This does not fill the excluded center or assign a division-by-zero default.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-punctured-quotient-value · candidate
98. Original Quotient Value at a Permitted Punctured Point
The original quotient (x²−(10)²)/(x−(10)) has punctured real domain x≠a. Evaluate it at the permitted x=13. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
23/1
The denominator is nonzero here, so the exact original quotient value is 23/1. This does not fill the excluded center or assign a division-by-zero default.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-punctured-quotient-value · candidate
99. Original Quotient Value at a Permitted Punctured Point
The original quotient (x²−(11)²)/(x−(11)) has punctured real domain x≠a. Evaluate it at the permitted x=7. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
18/1
The denominator is nonzero here, so the exact original quotient value is 18/1. This does not fill the excluded center or assign a division-by-zero default.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-punctured-quotient-value · candidate
100. Original Quotient Value at a Permitted Punctured Point
The original quotient (x²−(12)²)/(x−(12)) has punctured real domain x≠a. Evaluate it at the permitted x=13. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
25/1
The denominator is nonzero here, so the exact original quotient value is 25/1. This does not fill the excluded center or assign a division-by-zero default.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-density-limit-punctured-quotient-value · candidate
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