Question

The function of $f(x)=|x|+\frac{|x|}{x}$ is

• A. continuous at the origin
• B. discontinuous at the origin because $|x|$ is discontinuous there
• C. discontinuous at the origin because $\frac{|x|}{x}$ is discontinuous there
• D. discontinuous at the origin because both $|x|$ and $\frac{|x|}{x}$ are discontinuous are

Answer 1

Answer: (C)

$f(x)=|x|+\frac{|x|}{x}$
$LHL$ of $x=0$
$\displaystyle\lim _{x \rightarrow 0^{-}} f(x)=\displaystyle\lim _{x \rightarrow 0^{-}}-x+\frac{-x}{x}$
$=\displaystyle\lim _{x \rightarrow 0^{-}}-x-1=-1$
$\displaystyle\lim _{x \rightarrow 0^{+}} f(x)=\displaystyle\lim _{x \rightarrow 0^{+}}(x)+\frac{(x)}{x}$
$=\displaystyle\lim _{x \rightarrow 0^{+}} x+1=1$
$\therefore LHL \neq R H L$ at $x=0$
$\therefore f(x)$ is discontinuous at $x=0$