Question

The value of $\displaystyle\lim_{x\to0} \frac{\left|x\right|}{x}$ is

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

Answer 1

Answer: (D)

Does not exist

Answer 2

$LHL =\displaystyle\lim _{ h \rightarrow 0} f (0- h )=\displaystyle\lim _{ h \rightarrow 0} \frac{|0- h |}{0- h }=\displaystyle\lim _{ h \rightarrow 0} \frac{ h }{- h }=\displaystyle\lim _{ h \rightarrow 0}(-1)=-1$
$RHL =\displaystyle\lim _{ x \rightarrow 0} f (0+ h )=\displaystyle\lim _{ h \rightarrow 0} \frac{|0+ h |}{0+ h }=\displaystyle\lim _{ h \rightarrow 0} \frac{ h }{ h }=\displaystyle\lim _{ h \rightarrow 0}(1)=1$
$\therefore \,LHL \neq RHL$
$\Rightarrow \,\displaystyle\lim _{x \rightarrow 0} \frac{| x |}{ x }$ does not exist