# Concept derivation guides

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Concept guide



## [Function limits: a removable-factor quotient, affine δ bounds and unequal one-sided limits](/en/derivations/derivation-mko-function-limit-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-function-limit)

On punctured real domains: (1) prove by ε-δ that f(x)=(x²−9)/(x−3) has lim_(x→3) f(x)=6 while f(3) stays undefined; (2) for the affine g(x)=4x−1 at a=2 with ε=1/2, prove that δ=1/8 is valid (the equality |m|δ=ε is allowed) and δ=1/7 is not; (3) for the branch function h(x)=x+1 (x<1), h(x)=2x+3 (x>1), find the one-sided limits 2 and 5 and prove that the two-sided limit does not exist.

Function limits on punctured real domains

derivation-mko-function-limit-01





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