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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-20261008-r02-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.
On R∖{0}, f(x)=(0)x+(0). Find its punctured two-sided limit as x→0, without assigning f(0).
0
The affine approach value is m*a+b=0. The original function's punctured domain does not require a defined center value.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine · candidate
On R∖{0}, f(x)=(-2)x+(-11). Find its punctured two-sided limit as x→0, without assigning f(0).
-11
The affine approach value is m*a+b=-11. The original function's punctured domain does not require a defined center value.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine · candidate
On R∖{3}, f(x)=(3)x+(-10). Find its punctured two-sided limit as x→3, without assigning f(3).
-1
The affine approach value is m*a+b=-1. The original function's punctured domain does not require a defined center value.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine · candidate
On R∖{4}, f(x)=(-4)x+(-9). Find its punctured two-sided limit as x→4, without assigning f(4).
-25
The affine approach value is m*a+b=-25. The original function's punctured domain does not require a defined center value.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine · candidate
On R∖{1}, f(x)=(5)x+(-8). Find its punctured two-sided limit as x→1, without assigning f(1).
-3
The affine approach value is m*a+b=-3. The original function's punctured domain does not require a defined center value.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine · candidate
On R∖{2}, f(x)=(-6)x+(-7). Find its punctured two-sided limit as x→2, without assigning f(2).
-19
The affine approach value is m*a+b=-19. The original function's punctured domain does not require a defined center value.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine · candidate
On R∖{3}, f(x)=(7)x+(-6). Find its punctured two-sided limit as x→3, without assigning f(3).
15
The affine approach value is m*a+b=15. The original function's punctured domain does not require a defined center value.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine · candidate
On R∖{4}, f(x)=(-8)x+(-5). Find its punctured two-sided limit as x→4, without assigning f(4).
-37
The affine approach value is m*a+b=-37. The original function's punctured domain does not require a defined center value.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine · candidate
On R∖{1}, f(x)=(9)x+(-4). Find its punctured two-sided limit as x→1, without assigning f(1).
5
The affine approach value is m*a+b=5. The original function's punctured domain does not require a defined center value.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine · candidate
On R∖{2}, f(x)=(-10)x+(-3). Find its punctured two-sided limit as x→2, without assigning f(2).
-23
The affine approach value is m*a+b=-23. The original function's punctured domain does not require a defined center value.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine · candidate
On R∖{3}, f(x)=(11)x+(-2). Find its punctured two-sided limit as x→3, without assigning f(3).
31
The affine approach value is m*a+b=31. The original function's punctured domain does not require a defined center value.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine · candidate
On R∖{4}, f(x)=(-12)x+(-1). Find its punctured two-sided limit as x→4, without assigning f(4).
-49
The affine approach value is m*a+b=-49. The original function's punctured domain does not require a defined center value.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine · candidate
On R∖{1}, f(x)=(13)x+(0). Find its punctured two-sided limit as x→1, without assigning f(1).
13
The affine approach value is m*a+b=13. The original function's punctured domain does not require a defined center value.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine · candidate
On R∖{2}, f(x)=(-14)x+(1). Find its punctured two-sided limit as x→2, without assigning f(2).
-27
The affine approach value is m*a+b=-27. The original function's punctured domain does not require a defined center value.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine · candidate
On R∖{3}, f(x)=(15)x+(2). Find its punctured two-sided limit as x→3, without assigning f(3).
47
The affine approach value is m*a+b=47. The original function's punctured domain does not require a defined center value.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine · candidate
On R∖{4}, f(x)=(-16)x+(3). Find its punctured two-sided limit as x→4, without assigning f(4).
-61
The affine approach value is m*a+b=-61. The original function's punctured domain does not require a defined center value.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine · candidate
On R∖{1}, f(x)=(17)x+(4). Find its punctured two-sided limit as x→1, without assigning f(1).
21
The affine approach value is m*a+b=21. The original function's punctured domain does not require a defined center value.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine · candidate
On R∖{2}, f(x)=(-18)x+(5). Find its punctured two-sided limit as x→2, without assigning f(2).
-31
The affine approach value is m*a+b=-31. The original function's punctured domain does not require a defined center value.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine · candidate
On R∖{3}, f(x)=(19)x+(6). Find its punctured two-sided limit as x→3, without assigning f(3).
63
The affine approach value is m*a+b=63. The original function's punctured domain does not require a defined center value.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine · candidate
On R∖{4}, f(x)=(-20)x+(7). Find its punctured two-sided limit as x→4, without assigning f(4).
-73
The affine approach value is m*a+b=-73. The original function's punctured domain does not require a defined center value.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine · candidate
On R∖{1}, f(x)=(21)x+(8). Find its punctured two-sided limit as x→1, without assigning f(1).
29
The affine approach value is m*a+b=29. The original function's punctured domain does not require a defined center value.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine · candidate
On R∖{2}, f(x)=(-22)x+(9). Find its punctured two-sided limit as x→2, without assigning f(2).
-35
The affine approach value is m*a+b=-35. The original function's punctured domain does not require a defined center value.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine · candidate
On R∖{3}, f(x)=(23)x+(10). Find its punctured two-sided limit as x→3, without assigning f(3).
79
The affine approach value is m*a+b=79. The original function's punctured domain does not require a defined center value.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine · candidate
On R∖{4}, f(x)=(-24)x+(11). Find its punctured two-sided limit as x→4, without assigning f(4).
-85
The affine approach value is m*a+b=-85. The original function's punctured domain does not require a defined center value.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine · candidate
On R∖{1}, f(x)=(25)x+(12). Find its punctured two-sided limit as x→1, without assigning f(1).
37
The affine approach value is m*a+b=37. The original function's punctured domain does not require a defined center value.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine · candidate
On R∖{-12}, f(x)=(x²−(-12)²)/(x−(-12)). Find its punctured two-sided limit as x→-12.
-24
For x≠a, cancel the nonzero factor x−a to obtain x+a; its approach value is 2a=-24. This does not define the original quotient at the center or perform division by zero.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-removable-quotient · candidate
On R∖{-11}, f(x)=(x²−(-11)²)/(x−(-11)). Find its punctured two-sided limit as x→-11.
-22
For x≠a, cancel the nonzero factor x−a to obtain x+a; its approach value is 2a=-22. This does not define the original quotient at the center or perform division by zero.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-removable-quotient · candidate
On R∖{-10}, f(x)=(x²−(-10)²)/(x−(-10)). Find its punctured two-sided limit as x→-10.
-20
For x≠a, cancel the nonzero factor x−a to obtain x+a; its approach value is 2a=-20. This does not define the original quotient at the center or perform division by zero.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-removable-quotient · candidate
On R∖{-9}, f(x)=(x²−(-9)²)/(x−(-9)). Find its punctured two-sided limit as x→-9.
-18
For x≠a, cancel the nonzero factor x−a to obtain x+a; its approach value is 2a=-18. This does not define the original quotient at the center or perform division by zero.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-removable-quotient · candidate
On R∖{-8}, f(x)=(x²−(-8)²)/(x−(-8)). Find its punctured two-sided limit as x→-8.
-16
For x≠a, cancel the nonzero factor x−a to obtain x+a; its approach value is 2a=-16. This does not define the original quotient at the center or perform division by zero.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-removable-quotient · candidate
On R∖{-7}, f(x)=(x²−(-7)²)/(x−(-7)). Find its punctured two-sided limit as x→-7.
-14
For x≠a, cancel the nonzero factor x−a to obtain x+a; its approach value is 2a=-14. This does not define the original quotient at the center or perform division by zero.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-removable-quotient · candidate
On R∖{-6}, f(x)=(x²−(-6)²)/(x−(-6)). Find its punctured two-sided limit as x→-6.
-12
For x≠a, cancel the nonzero factor x−a to obtain x+a; its approach value is 2a=-12. This does not define the original quotient at the center or perform division by zero.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-removable-quotient · candidate
On R∖{-5}, f(x)=(x²−(-5)²)/(x−(-5)). Find its punctured two-sided limit as x→-5.
-10
For x≠a, cancel the nonzero factor x−a to obtain x+a; its approach value is 2a=-10. This does not define the original quotient at the center or perform division by zero.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-removable-quotient · candidate
On R∖{-4}, f(x)=(x²−(-4)²)/(x−(-4)). Find its punctured two-sided limit as x→-4.
-8
For x≠a, cancel the nonzero factor x−a to obtain x+a; its approach value is 2a=-8. This does not define the original quotient at the center or perform division by zero.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-removable-quotient · candidate
On R∖{-3}, f(x)=(x²−(-3)²)/(x−(-3)). Find its punctured two-sided limit as x→-3.
-6
For x≠a, cancel the nonzero factor x−a to obtain x+a; its approach value is 2a=-6. This does not define the original quotient at the center or perform division by zero.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-removable-quotient · candidate
On R∖{-2}, f(x)=(x²−(-2)²)/(x−(-2)). Find its punctured two-sided limit as x→-2.
-4
For x≠a, cancel the nonzero factor x−a to obtain x+a; its approach value is 2a=-4. This does not define the original quotient at the center or perform division by zero.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-removable-quotient · candidate
On R∖{-1}, f(x)=(x²−(-1)²)/(x−(-1)). Find its punctured two-sided limit as x→-1.
-2
For x≠a, cancel the nonzero factor x−a to obtain x+a; its approach value is 2a=-2. This does not define the original quotient at the center or perform division by zero.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-removable-quotient · candidate
On R∖{0}, f(x)=(x²−(0)²)/(x−(0)). Find its punctured two-sided limit as x→0.
0
For x≠a, cancel the nonzero factor x−a to obtain x+a; its approach value is 2a=0. This does not define the original quotient at the center or perform division by zero.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-removable-quotient · candidate
On R∖{1}, f(x)=(x²−(1)²)/(x−(1)). Find its punctured two-sided limit as x→1.
2
For x≠a, cancel the nonzero factor x−a to obtain x+a; its approach value is 2a=2. This does not define the original quotient at the center or perform division by zero.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-removable-quotient · candidate
On R∖{2}, f(x)=(x²−(2)²)/(x−(2)). Find its punctured two-sided limit as x→2.
4
For x≠a, cancel the nonzero factor x−a to obtain x+a; its approach value is 2a=4. This does not define the original quotient at the center or perform division by zero.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-removable-quotient · candidate
On R∖{3}, f(x)=(x²−(3)²)/(x−(3)). Find its punctured two-sided limit as x→3.
6
For x≠a, cancel the nonzero factor x−a to obtain x+a; its approach value is 2a=6. This does not define the original quotient at the center or perform division by zero.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-removable-quotient · candidate
On R∖{4}, f(x)=(x²−(4)²)/(x−(4)). Find its punctured two-sided limit as x→4.
8
For x≠a, cancel the nonzero factor x−a to obtain x+a; its approach value is 2a=8. This does not define the original quotient at the center or perform division by zero.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-removable-quotient · candidate
On R∖{5}, f(x)=(x²−(5)²)/(x−(5)). Find its punctured two-sided limit as x→5.
10
For x≠a, cancel the nonzero factor x−a to obtain x+a; its approach value is 2a=10. This does not define the original quotient at the center or perform division by zero.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-removable-quotient · candidate
On R∖{6}, f(x)=(x²−(6)²)/(x−(6)). Find its punctured two-sided limit as x→6.
12
For x≠a, cancel the nonzero factor x−a to obtain x+a; its approach value is 2a=12. This does not define the original quotient at the center or perform division by zero.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-removable-quotient · candidate
On R∖{7}, f(x)=(x²−(7)²)/(x−(7)). Find its punctured two-sided limit as x→7.
14
For x≠a, cancel the nonzero factor x−a to obtain x+a; its approach value is 2a=14. This does not define the original quotient at the center or perform division by zero.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-removable-quotient · candidate
On R∖{8}, f(x)=(x²−(8)²)/(x−(8)). Find its punctured two-sided limit as x→8.
16
For x≠a, cancel the nonzero factor x−a to obtain x+a; its approach value is 2a=16. This does not define the original quotient at the center or perform division by zero.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-removable-quotient · candidate
On R∖{9}, f(x)=(x²−(9)²)/(x−(9)). Find its punctured two-sided limit as x→9.
18
For x≠a, cancel the nonzero factor x−a to obtain x+a; its approach value is 2a=18. This does not define the original quotient at the center or perform division by zero.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-removable-quotient · candidate
On R∖{10}, f(x)=(x²−(10)²)/(x−(10)). Find its punctured two-sided limit as x→10.
20
For x≠a, cancel the nonzero factor x−a to obtain x+a; its approach value is 2a=20. This does not define the original quotient at the center or perform division by zero.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-removable-quotient · candidate
On R∖{11}, f(x)=(x²−(11)²)/(x−(11)). Find its punctured two-sided limit as x→11.
22
For x≠a, cancel the nonzero factor x−a to obtain x+a; its approach value is 2a=22. This does not define the original quotient at the center or perform division by zero.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-removable-quotient · candidate
On R∖{12}, f(x)=(x²−(12)²)/(x−(12)). Find its punctured two-sided limit as x→12.
24
For x≠a, cancel the nonzero factor x−a to obtain x+a; its approach value is 2a=24. This does not define the original quotient at the center or perform division by zero.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-removable-quotient · candidate
For arbitrary real A,L, define f(x)=L+(0)(x−A) for x≠A on R∖{A}. With ε=1/1>0 and δ=5/1>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
Yes
Check |m|δ≤ε by exact cross products, giving Yes. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A).
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine-delta · candidate
For arbitrary real A,L, define f(x)=L+(-1)(x−A) for x≠A on R∖{A}. With ε=2/2>0 and δ=2/2>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
Yes
Check |m|δ≤ε by exact cross products, giving Yes. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A).
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine-delta · candidate
For arbitrary real A,L, define f(x)=L+(2)(x−A) for x≠A on R∖{A}. With ε=3/3>0 and δ=3/12>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
Yes
Check |m|δ≤ε by exact cross products, giving Yes. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A).
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine-delta · candidate
For arbitrary real A,L, define f(x)=L+(-3)(x−A) for x≠A on R∖{A}. With ε=1/4>0 and δ=2/12>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
No
Check |m|δ≤ε by exact cross products, giving No. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A).
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine-delta · candidate
For arbitrary real A,L, define f(x)=L+(4)(x−A) for x≠A on R∖{A}. With ε=2/1>0 and δ=2/4>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
Yes
Check |m|δ≤ε by exact cross products, giving Yes. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A).
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine-delta · candidate
For arbitrary real A,L, define f(x)=L+(-5)(x−A) for x≠A on R∖{A}. With ε=3/2>0 and δ=3/20>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
Yes
Check |m|δ≤ε by exact cross products, giving Yes. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A).
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine-delta · candidate
For arbitrary real A,L, define f(x)=L+(6)(x−A) for x≠A on R∖{A}. With ε=1/3>0 and δ=2/18>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
No
Check |m|δ≤ε by exact cross products, giving No. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A).
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine-delta · candidate
For arbitrary real A,L, define f(x)=L+(-7)(x−A) for x≠A on R∖{A}. With ε=2/4>0 and δ=2/28>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
Yes
Check |m|δ≤ε by exact cross products, giving Yes. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A).
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine-delta · candidate
For arbitrary real A,L, define f(x)=L+(8)(x−A) for x≠A on R∖{A}. With ε=3/1>0 and δ=3/16>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
Yes
Check |m|δ≤ε by exact cross products, giving Yes. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A).
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine-delta · candidate
For arbitrary real A,L, define f(x)=L+(-9)(x−A) for x≠A on R∖{A}. With ε=1/2>0 and δ=2/18>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
No
Check |m|δ≤ε by exact cross products, giving No. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A).
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine-delta · candidate
For arbitrary real A,L, define f(x)=L+(10)(x−A) for x≠A on R∖{A}. With ε=2/3>0 and δ=2/30>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
Yes
Check |m|δ≤ε by exact cross products, giving Yes. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A).
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine-delta · candidate
For arbitrary real A,L, define f(x)=L+(-11)(x−A) for x≠A on R∖{A}. With ε=3/4>0 and δ=3/88>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
Yes
Check |m|δ≤ε by exact cross products, giving Yes. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A).
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine-delta · candidate
For arbitrary real A,L, define f(x)=L+(12)(x−A) for x≠A on R∖{A}. With ε=1/1>0 and δ=2/12>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
No
Check |m|δ≤ε by exact cross products, giving No. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A).
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine-delta · candidate
For arbitrary real A,L, define f(x)=L+(-13)(x−A) for x≠A on R∖{A}. With ε=2/2>0 and δ=2/26>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
Yes
Check |m|δ≤ε by exact cross products, giving Yes. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A).
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine-delta · candidate
For arbitrary real A,L, define f(x)=L+(14)(x−A) for x≠A on R∖{A}. With ε=3/3>0 and δ=3/84>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
Yes
Check |m|δ≤ε by exact cross products, giving Yes. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A).
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine-delta · candidate
For arbitrary real A,L, define f(x)=L+(-15)(x−A) for x≠A on R∖{A}. With ε=1/4>0 and δ=2/60>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
No
Check |m|δ≤ε by exact cross products, giving No. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A).
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine-delta · candidate
For arbitrary real A,L, define f(x)=L+(16)(x−A) for x≠A on R∖{A}. With ε=2/1>0 and δ=2/16>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
Yes
Check |m|δ≤ε by exact cross products, giving Yes. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A).
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine-delta · candidate
For arbitrary real A,L, define f(x)=L+(-17)(x−A) for x≠A on R∖{A}. With ε=3/2>0 and δ=3/68>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
Yes
Check |m|δ≤ε by exact cross products, giving Yes. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A).
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine-delta · candidate
For arbitrary real A,L, define f(x)=L+(18)(x−A) for x≠A on R∖{A}. With ε=1/3>0 and δ=2/54>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
No
Check |m|δ≤ε by exact cross products, giving No. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A).
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine-delta · candidate
For arbitrary real A,L, define f(x)=L+(-19)(x−A) for x≠A on R∖{A}. With ε=2/4>0 and δ=2/76>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
Yes
Check |m|δ≤ε by exact cross products, giving Yes. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A).
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine-delta · candidate
For arbitrary real A,L, define f(x)=L+(20)(x−A) for x≠A on R∖{A}. With ε=3/1>0 and δ=3/40>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
Yes
Check |m|δ≤ε by exact cross products, giving Yes. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A).
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine-delta · candidate
For arbitrary real A,L, define f(x)=L+(-21)(x−A) for x≠A on R∖{A}. With ε=1/2>0 and δ=2/42>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
No
Check |m|δ≤ε by exact cross products, giving No. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A).
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine-delta · candidate
For arbitrary real A,L, define f(x)=L+(22)(x−A) for x≠A on R∖{A}. With ε=2/3>0 and δ=2/66>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
Yes
Check |m|δ≤ε by exact cross products, giving Yes. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A).
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine-delta · candidate
For arbitrary real A,L, define f(x)=L+(-23)(x−A) for x≠A on R∖{A}. With ε=3/4>0 and δ=3/184>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
Yes
Check |m|δ≤ε by exact cross products, giving Yes. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A).
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine-delta · candidate
For arbitrary real A,L, define f(x)=L+(24)(x−A) for x≠A on R∖{A}. With ε=1/1>0 and δ=2/24>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?
No
Check |m|δ≤ε by exact cross products, giving No. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A).
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-affine-delta · candidate
On the punctured domain R∖{-12}, The left branch at x<-12 is (1)x+(0); the right branch at x>-12 is (-1)x+(-24). Find the finite two-sided limit, or answer does not exist when the two side limits differ.
-12
The left and right limits are [-12, -12]. They must agree, giving -12; nonexistence of a two-sided limit does not erase either one-sided limit.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-two-sided-affine · candidate
On the punctured domain R∖{-11}, The left branch at x<-11 is (2)x+(11); the right branch at x>-11 is (-2)x+(-20). Find the finite two-sided limit, or answer does not exist when the two side limits differ.
does not exist
The left and right limits are [-11, 2]. They must agree, giving does not exist; nonexistence of a two-sided limit does not erase either one-sided limit.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-two-sided-affine · candidate
On the punctured domain R∖{-10}, The left branch at x<-10 is (3)x+(20); the right branch at x>-10 is (-3)x+(-27). Find the finite two-sided limit, or answer does not exist when the two side limits differ.
does not exist
The left and right limits are [-10, 3]. They must agree, giving does not exist; nonexistence of a two-sided limit does not erase either one-sided limit.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-two-sided-affine · candidate
On the punctured domain R∖{-9}, The left branch at x<-9 is (4)x+(27); the right branch at x>-9 is (-1)x+(-18). Find the finite two-sided limit, or answer does not exist when the two side limits differ.
-9
The left and right limits are [-9, -9]. They must agree, giving -9; nonexistence of a two-sided limit does not erase either one-sided limit.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-two-sided-affine · candidate
On the punctured domain R∖{-8}, The left branch at x<-8 is (1)x+(0); the right branch at x>-8 is (-2)x+(-11). Find the finite two-sided limit, or answer does not exist when the two side limits differ.
does not exist
The left and right limits are [-8, 5]. They must agree, giving does not exist; nonexistence of a two-sided limit does not erase either one-sided limit.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-two-sided-affine · candidate
On the punctured domain R∖{-7}, The left branch at x<-7 is (2)x+(7); the right branch at x>-7 is (-3)x+(-15). Find the finite two-sided limit, or answer does not exist when the two side limits differ.
does not exist
The left and right limits are [-7, 6]. They must agree, giving does not exist; nonexistence of a two-sided limit does not erase either one-sided limit.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-two-sided-affine · candidate
On the punctured domain R∖{-6}, The left branch at x<-6 is (3)x+(12); the right branch at x>-6 is (-1)x+(-12). Find the finite two-sided limit, or answer does not exist when the two side limits differ.
-6
The left and right limits are [-6, -6]. They must agree, giving -6; nonexistence of a two-sided limit does not erase either one-sided limit.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-two-sided-affine · candidate
On the punctured domain R∖{-5}, The left branch at x<-5 is (4)x+(15); the right branch at x>-5 is (-2)x+(-2). Find the finite two-sided limit, or answer does not exist when the two side limits differ.
does not exist
The left and right limits are [-5, 8]. They must agree, giving does not exist; nonexistence of a two-sided limit does not erase either one-sided limit.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-two-sided-affine · candidate
On the punctured domain R∖{-4}, The left branch at x<-4 is (1)x+(0); the right branch at x>-4 is (-3)x+(-3). Find the finite two-sided limit, or answer does not exist when the two side limits differ.
does not exist
The left and right limits are [-4, 9]. They must agree, giving does not exist; nonexistence of a two-sided limit does not erase either one-sided limit.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-two-sided-affine · candidate
On the punctured domain R∖{-3}, The left branch at x<-3 is (2)x+(3); the right branch at x>-3 is (-1)x+(-6). Find the finite two-sided limit, or answer does not exist when the two side limits differ.
-3
The left and right limits are [-3, -3]. They must agree, giving -3; nonexistence of a two-sided limit does not erase either one-sided limit.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-two-sided-affine · candidate
On the punctured domain R∖{-2}, The left branch at x<-2 is (3)x+(4); the right branch at x>-2 is (-2)x+(7). Find the finite two-sided limit, or answer does not exist when the two side limits differ.
does not exist
The left and right limits are [-2, 11]. They must agree, giving does not exist; nonexistence of a two-sided limit does not erase either one-sided limit.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-two-sided-affine · candidate
On the punctured domain R∖{-1}, The left branch at x<-1 is (4)x+(3); the right branch at x>-1 is (-3)x+(9). Find the finite two-sided limit, or answer does not exist when the two side limits differ.
does not exist
The left and right limits are [-1, 12]. They must agree, giving does not exist; nonexistence of a two-sided limit does not erase either one-sided limit.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-two-sided-affine · candidate
On the punctured domain R∖{0}, The left branch at x<0 is (1)x+(0); the right branch at x>0 is (-1)x+(0). Find the finite two-sided limit, or answer does not exist when the two side limits differ.
0
The left and right limits are [0, 0]. They must agree, giving 0; nonexistence of a two-sided limit does not erase either one-sided limit.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-two-sided-affine · candidate
On the punctured domain R∖{1}, The left branch at x<1 is (2)x+(-1); the right branch at x>1 is (-2)x+(16). Find the finite two-sided limit, or answer does not exist when the two side limits differ.
does not exist
The left and right limits are [1, 14]. They must agree, giving does not exist; nonexistence of a two-sided limit does not erase either one-sided limit.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-two-sided-affine · candidate
On the punctured domain R∖{2}, The left branch at x<2 is (3)x+(-4); the right branch at x>2 is (-3)x+(21). Find the finite two-sided limit, or answer does not exist when the two side limits differ.
does not exist
The left and right limits are [2, 15]. They must agree, giving does not exist; nonexistence of a two-sided limit does not erase either one-sided limit.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-two-sided-affine · candidate
On the punctured domain R∖{3}, The left branch at x<3 is (4)x+(-9); the right branch at x>3 is (-1)x+(6). Find the finite two-sided limit, or answer does not exist when the two side limits differ.
3
The left and right limits are [3, 3]. They must agree, giving 3; nonexistence of a two-sided limit does not erase either one-sided limit.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-two-sided-affine · candidate
On the punctured domain R∖{4}, The left branch at x<4 is (1)x+(0); the right branch at x>4 is (-2)x+(25). Find the finite two-sided limit, or answer does not exist when the two side limits differ.
does not exist
The left and right limits are [4, 17]. They must agree, giving does not exist; nonexistence of a two-sided limit does not erase either one-sided limit.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-two-sided-affine · candidate
On the punctured domain R∖{5}, The left branch at x<5 is (2)x+(-5); the right branch at x>5 is (-3)x+(33). Find the finite two-sided limit, or answer does not exist when the two side limits differ.
does not exist
The left and right limits are [5, 18]. They must agree, giving does not exist; nonexistence of a two-sided limit does not erase either one-sided limit.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-two-sided-affine · candidate
On the punctured domain R∖{6}, The left branch at x<6 is (3)x+(-12); the right branch at x>6 is (-1)x+(12). Find the finite two-sided limit, or answer does not exist when the two side limits differ.
6
The left and right limits are [6, 6]. They must agree, giving 6; nonexistence of a two-sided limit does not erase either one-sided limit.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-two-sided-affine · candidate
On the punctured domain R∖{7}, The left branch at x<7 is (4)x+(-21); the right branch at x>7 is (-2)x+(34). Find the finite two-sided limit, or answer does not exist when the two side limits differ.
does not exist
The left and right limits are [7, 20]. They must agree, giving does not exist; nonexistence of a two-sided limit does not erase either one-sided limit.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-two-sided-affine · candidate
On the punctured domain R∖{8}, The left branch at x<8 is (1)x+(0); the right branch at x>8 is (-3)x+(45). Find the finite two-sided limit, or answer does not exist when the two side limits differ.
does not exist
The left and right limits are [8, 21]. They must agree, giving does not exist; nonexistence of a two-sided limit does not erase either one-sided limit.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-two-sided-affine · candidate
On the punctured domain R∖{9}, The left branch at x<9 is (2)x+(-9); the right branch at x>9 is (-1)x+(18). Find the finite two-sided limit, or answer does not exist when the two side limits differ.
9
The left and right limits are [9, 9]. They must agree, giving 9; nonexistence of a two-sided limit does not erase either one-sided limit.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-two-sided-affine · candidate
On the punctured domain R∖{10}, The left branch at x<10 is (3)x+(-20); the right branch at x>10 is (-2)x+(43). Find the finite two-sided limit, or answer does not exist when the two side limits differ.
does not exist
The left and right limits are [10, 23]. They must agree, giving does not exist; nonexistence of a two-sided limit does not erase either one-sided limit.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-two-sided-affine · candidate
On the punctured domain R∖{11}, The left branch at x<11 is (4)x+(-33); the right branch at x>11 is (-3)x+(57). Find the finite two-sided limit, or answer does not exist when the two side limits differ.
does not exist
The left and right limits are [11, 24]. They must agree, giving does not exist; nonexistence of a two-sided limit does not erase either one-sided limit.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-two-sided-affine · candidate
On the punctured domain R∖{12}, The left branch at x<12 is (1)x+(0); the right branch at x>12 is (-1)x+(24). Find the finite two-sided limit, or answer does not exist when the two side limits differ.
12
The left and right limits are [12, 12]. They must agree, giving 12; nonexistence of a two-sided limit does not erase either one-sided limit.
Reviewed step-by-step derivation
Function limits on punctured real domains · qf-function-limit-two-sided-affine · candidate
Canonical question JSONL · English JSONL · Validation report