Test the continuity of the following function at the origin: `f(x)={(|x|)/x ;x!=0`
, `1;x=0`

Answer

Hence,`f(x)` is discontinuous at origin.

Explanation

ATQ,
$$ f(x)= \begin{cases}\frac{x}{|x|}, & x \ne 0 \\ 1, & x=0\end{cases} $$
LHL at `x=0`,
$$ \lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0} f(0-h)=\lim _{h \rightarrow 0} f(-h) $$
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