Question:
Show that $\lim _{x \rightarrow 0} \frac{x}{|x|}$ does not exist.
Solution:
Left Hand Limit(L.H.L.):
$\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{-}} \frac{x}{|x|}$
$=\lim _{x \rightarrow 0^{-}} \frac{x}{(-x)}$
$=\lim _{x \rightarrow 0^{-}}-1$
$=-1$
Right Hand Limit(R.H.L.):
$\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0^{+}} \frac{x}{|x|}$
$=\lim _{x \rightarrow 0^{+}} \frac{x}{(x)}$
$=\lim _{x \rightarrow 0^{+}} 1$
$=1$
Since
$\lim _{x \rightarrow 0^{-}} f(x) \neq \lim _{x \rightarrow 0^{+}} f(x)$
Thus, $\lim _{x \rightarrow 0} \frac{x}{|x|}$ does not exist.
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