Question

explain why the limit does not exist.
lim_{x\to0}\frac{x}{|x|}
select the correct choice below and fill in the answer boxes within your choice.
a. as x approaches 0 from the left, \frac{x}{|x|} approaches \square. as x approaches 0 from the right, \frac{x}{|x|} approaches \square. this means there is no single number l that the
function values get arbitrarily close to as x\to0.
b. the function is not defined at x = 0, so the limit cannot exist.

Explanation:

Step1: Recall the definition of absolute value

For \(x<0\), \(|x|=-x\). So when \(x\to0^{-}\) (approaching \(0\) from the left), \(\frac{x}{|x|}=\frac{x}{-x}=- 1\).

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

For \(x > 0\), \(|x|=x\). So when \(x\to0^{+}\) (approaching \(0\) from the right), \(\frac{x}{|x|}=\frac{x}{x}=1\).

Answer:

A. As \(x\) approaches \(0\) from the left, \(\frac{x}{|x|}\) approaches \(-1\). As \(x\) approaches \(0\) from the right, \(\frac{x}{|x|}\) approaches \(1\). This means there is no single number \(L\) that the function values get arbitrarily close to as \(x\to0\).