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.