Question

Evaluate $\lim_{x\rightarrow 0}$f(x), where { |x|/x, x≠0 0, x=0

Answer 1

The given function is
f(x) ={ $\frac{|x|}{x}$, x≠0 0, x=0
$\lim_{x\rightarrow 0^-}$ f(x) = $\lim_{x\rightarrow 0^-}$[$\frac{|x|}{x}$]
= $\lim_{x\rightarrow 0}$ ($\frac{-x}{x}$) [When x is negative, |x| = -x]
= $\lim_{x\rightarrow 0}$(-1)
= -1
$\lim_{x\rightarrow 0^+}$f(x) = $\lim_{x\rightarrow 0^+}$ [$\frac{|x|}{x}$]
= $\lim_{x\rightarrow 0}$ ($\frac{x}{x}$) [When x is Positive, |x| = x]
= $\lim_{x\rightarrow 0}$ (1)
= 1
It is observed that $\lim_{x\rightarrow 0^-}$ f(x)≠$\lim_{x\rightarrow 0^+}$f(x)
Hence, $\lim_{x\rightarrow 0}$ does not exist.