Chapter 1.5, Problem 35ES

A function $f$ is said to have a removable discontinuity at $x=c$ if ${\lim }_{x\to c}f\left(x\right)$ exists but $f$ is not continuous at $x=c,$ either because $f$ is not defined at $c$ or because the definition for $f\left(c\right)$ differs from the value of the limit. This terminology will be needed in these exercises.

Find the values of $x$ (if any) at which $f$ is not continuous, and determine whether each such value is a removable discontinuity.

(a) $f\left(x\right)=\frac{\left|x\right|}{x}$

(b) $f\left(x\right)=\frac{x^2+3x}{x+3}$

(c) $f\left(x\right)=\frac{x-2}{\left|x\right|-2}$