Q. The function $f(x) = |x| + \frac{|x|}{x}$ is
A) Discontinuous at origin because $|x|$ is discontinuous there
B) Continuous at origin
C) Discontinuous at origin because both $|x|$ and $\frac{|x|}{x}$ are discontinuous there
D) Discontinuous at the origin because $\frac{|x|}{x}$ is discontinuous there
Explanation
(d)
Given, $f(x) = |x| + \frac{|x|}{x}$
Let $f_{1}(x) = |x|,\ \ f_{2}(x) = \frac{|x|}{x}$
LHL$= \lim_{x{\rightarrow 0}^{-}}{f_{1}(x) =}\lim_{x{\rightarrow 0}^{-}}|x| = 0$
And RHL$\lim_{x{\rightarrow 0}^{+}}f_{1}(x) = \lim_{x{\rightarrow 0}^{+}}|x| = 0$
Here, LHL=RHL=$f(0),\ f_{1}(x)$ is continuous
LHL$= \lim_{x{\rightarrow 0}^{-}}\frac{|x|}{x} = \lim_{h \rightarrow 0}\frac{|0 - h|}{0 - h} = - 1$
RHL$= \lim_{x{\rightarrow 0}^{+}}\frac{|x|}{x} = \lim_{h \rightarrow 0}\frac{|0 + h|}{h} = 1$
$\therefore$ LHL$\neq$RHL, $f_{2}(x)$ is discontinuous
Hence, $f(x)$ is discontinuous at $x = 0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.