Question

Find the value of $\underset{x\to 0}{lim}\frac{|x|}{x}$

• A. 1
• B. -1
• C. 0
• D. Does not exist

Answer 1

The correct option is D

Does not exist

$\frac{|x|}{x}$ is the signum function without the point (0,0). Let's try to remove the modulus and simplify the

function.

$\frac{|x|}{x}=\frac{x}{x}$ if x > 0

$=1$

$\frac{|x|}{x}=\frac{-x}{x}$ if x < 0

$=-1$

At $x=0,\frac{|x|}{x}$ is not defined. So the garph will look like

we can see that as x approaches zero from left, the value of the function is -1, if the approaches from

right, the value is 1. So the limit does not exists at x = 0 even though R.H.L and L.H.L exists finitely.