Solve this

Question:

Let $f(x)=\left\{\begin{array}{l}\frac{x}{|x|^{\prime}} x \neq 0 \\ 0, x=0\end{array}\right.$

Show that $\lim _{x \rightarrow 0} f(x)$ 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 $\lim _{x \rightarrow 0^{-}} f(x) \neq \lim _{x \rightarrow 0^{+}} f(x), \lim _{x \rightarrow 0} f(x)$ does not exist

 

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