Question

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

Answer 1

The given function is
f(x), where { $\frac{x}{|x|}$, 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<0, |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 > 0, |x| = x]
= lim x →0 (1)
= 1
It is observed that lim x →0- f(x)≠ $\lim_{x\rightarrow 0^+}$f(x)
Hence, $\lim_{x\rightarrow 0}$ f(x) does not exist.