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.