Question

The value of $\underset{x\to 0}{lim}(\frac{1}{x^2}-cotx)$ is

• A. 0
• B. 1
• C. $\frac{1}{4}$
• D. none of these

Answer 1

The correct option is D none of these

${lim}_{x\to 0}(\frac{1}{x^2}-cotx)[\infty -\infty form]{lim}_{x\to 0}(\frac{1}{x^2}-\frac{\cos x}{\sin x}){lim}_{x\to 0}\left(\frac{\sin x-x^2\cos x}{x^2\sin x}\right)\Rightarrow \left(\frac{\sin x-x^2\cos x}{x^3}\right).\left(\frac{x}{\sin x}\right)\Rightarrow {lim}_{x\to 0}\left(\frac{\sin x-x^2\cos x}{x^3}\right)\times {lim}_{x\to 0}\left(\frac{x}{\sin x}\right)$

Using LHR.

$\Rightarrow {lim}_{x\to 0}\left(\frac{\cos x-(2x\cos x-x^2\sin x)}{3x^2}\right).1\Rightarrow {lim}_{x\to 0}\left(\frac{1-0}{0}\right)=\infty D)Ans.$