Question

The value of $\underset{x\to 0}{\lim }\frac{\left|x\right|}{x}$ is

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

Answer 1

Answer: (D)

$LHL=\underset{h\to 0}{\lim }f(0-h)=\underset{h\to 0}{\lim }\frac{|0-h|}{0-h}=\underset{h\to 0}{\lim }\frac{h}{-h}=\underset{h\to 0}{\lim }(-1)=-1$
$RHL=\underset{x\to 0}{\lim }f(0+h)=\underset{h\to 0}{\lim }\frac{|0+h|}{0+h}=\underset{h\to 0}{\lim }\frac{h}{h}=\underset{h\to 0}{\lim }(1)=1$
$\therefore LHL\ne RHL$
$\Rightarrow \underset{x\to 0}{\lim }\frac{|x|}{x}$ does not exist