Question

evaluate the given limits of the piece - wise defined function.
$f(x)=\

$$\begin{cases}\\frac{|x|}{x}&x\ eq0\\\\0&x = 0\\end{cases}$$

$
if you need to enter $\infty$, you may type infinity, or just inf. if the limit does not exist, you may type does not exist, or just dne.
a. $\lim_{x\to0^{-}}f(x)$
dne
b. $\lim_{x\to0^{+}}f(x)$
dne

Explanation:

Step1: Recall the definition of absolute - value

For \(x\lt0\), \(|x|=-x\). So when \(x\to0^-\), \(f(x)=\frac{|x|}{x}=\frac{-x}{x}=- 1\).

Step2: Recall the definition of absolute - value for positive \(x\)

For \(x\gt0\), \(|x| = x\). So when \(x\to0^+\), \(f(x)=\frac{|x|}{x}=\frac{x}{x}=1\).

Answer:

a. \(-1\)
b. \(1\)