Chapter 2.7, Problem 41E

Precise definitions for left- and right-sided limits Use the following definitions.

Assume f exists for all x near a with x > a. We say the limit of f(x) as x approaches a from the right of a is L and write $\underset{x\to a^{+}}{\lim }f(x)=L$, if for any ε > 0 there exists δ > 0 such that

$|f(x)-L|<\epsilon whenever0<x-a<\delta .$

Assume f exists for all x near a with x < a. We say the limit of f(x) as x approaches a from the left of a is L and write $\underset{x\to a^{-}}{\lim }f(x)=L$, if for any ε > 0 there exists δ > 0 such that

$|f(x)-L|<\epsilon whenever0<a-x<\delta .$

41. One-sided limit proofs Prove the following limits for

$f(x)=\{\begin{array}{cc}3x-4& \text{if }x<0\\ 2x-4& \text{if }x\ge 0.\end{array}$

1. a. $\underset{x\to 0^{+}}{\lim }f(x)=-4$
2. b. $\underset{x\to 0}{\lim }f(x)=-4$
3. c. $\underset{x\to 0^{-}}{\lim }f(x)=-4$