Chapter 2.3, Problem 116E

Proof

(a) Prove that if $\underset{x\to c}{\lim }\left|f\left(x\right)\right|=0$, then $\underset{x\to c}{\lim }f\left(x\right)=0$.

(Note: This is the converse of Exercise 114.)

(b) Prove that if $\underset{x\to c}{\lim }f\left(x\right)=L$, then $\underset{x\to c}{\lim }\left|f\left(x\right)\right|=\left|L\right|$.

[Hint: Use the inequality $\Vert f\left(x\right)|-|L\Vert \le \left|f\left(x\right)-L\right|$.]