Question

The function $f\left(x\right)=\left|x\right|+\frac{\left|x\right|}{x}$ is:

• A. discountinuous 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} $ discontinuous there.

Answer 1

Answer: (D)

Let $f\left(x\right)=\left|x\right|+\frac{\left|x\right|}{x}$
Let us consider
$f(x) = f_1(x) + f_2(x)$
Where $f_1(x) = |x|$ and $f_2 (x) = \frac{|x|}{x}$
Now at x = 0
$f_1 (0) = 0$
LHL = $ \displaystyle\lim_{x \to 0^-} |x|=0$ and
RHL $= \displaystyle\lim_{x \to 0^+} |x| = 0$
$ \therefore \:\:\: f_1(x)$ is continuous at $x = 0$
Now consider $f_2(x) = \frac{|x|}{x}$
Now LHL = $ \displaystyle\lim_{x \to 0^-} \frac{|x|}{x}$
Put $x = 0 - h$
$ \displaystyle\lim_{x \to 0^-} \frac{| 0 -h|}{ 0-h} = \displaystyle\lim_{x \to 0^-} - \frac{h}{h} = - 1$
Now RHL = $ \displaystyle\lim_{x \to 0^+} \frac{|x|}{x}$
$= \displaystyle\lim_{x \to 0^+} \frac{| 0 + h |}{ 0 + h}$ [By putting x = 0 + h]
$ = \displaystyle\lim_{x \to 0} \frac{ h}{ h} = 1$
Since LHL $\neq$ RHL
$\therefore \:\: f_2(x)$ is discontinuous at origin.
$\therefore \:\: f\left(x\right)=\left|x\right|+\frac{\left|x\right|}{x}$ is discontinuous at origin because $\frac{\left|x\right|}{x}$ is discontinuous.