Question

Which of the following function(s) not defined at $x=0$ has/have irremovable discontinuity at $x=0$?

• A. $f\left(x\right)=\frac{1}{\ln |x|}$
• B. $f\left(x\right)=\cos \left(\frac{|\sin x|}{x}\right)$
• C. $f\left(x\right)=x\sin \left(\frac{\pi }{x}\right)$
• D. $f\left(x\right)=\frac{1}{1+2^{\cot x}}$

Answer 1

The correct option is D $f\left(x\right)=\frac{1}{1+2^{\cot x}}$

* $\underset{x\to 0^{+}}{\text{lim}}\frac{1}{\ln |x|}=\underset{x\to 0^{-}}{\text{lim}}\frac{1}{\ln |x|}=0$

* $\underset{x\to 0^{+}}{\text{lim}}\cos \left(\frac{|\sin x|}{x}\right)=\underset{x\to 0^{-}}{\text{lim}}\cos \left(\frac{|\sin x|}{x}\right)=\cos 1$

* $\underset{x\to 0^{+}}{\text{lim}}x\sin \left(\frac{\pi }{x}\right)=\underset{x\to 0^{-}}{\text{lim}}x\sin \left(\frac{\pi }{x}\right)=0$

* $\underset{x\to 0^{+}}{\text{lim}}\frac{1}{1+2^{\cot x}}=0,\underset{x\to 0^{-}}{\text{lim}}\frac{1}{1+2^{\cot x}}=1$