Chapter 2.7, Problem 42E

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

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

42. Determining values of δ from a graph The function f in the figure satisfies $\underset{x\to 2^{+}}{\lim }f(x)=0$ and $\underset{x\to 2^{-}}{\lim }f(x)=1$. Determine all values of δ > 0 that satisfy each statement

1. a. |f(x) − 0| < 2 whenever 0 < x − 2 < δ
2. b. |f(x) − 0| < 1 whenever 0 < x − 2 < δ
3. c. |f(x) − 1| < 2 whenever 0 < 2 − x < δ
4. d. |f(x) − 1| < 1 whenever 0 < 2 − x < δ