Question

The value of $\underset{x\to 0}{lim}(\frac{1}{x}-\cot x)$ equals

• A. $1$
• B. $-1$
• C. $2$
• D. $0$

Answer 1

Answer: (D)

$0$

$\underset{x\to 0}{lim}\left(\frac{1}{x}-\cot x\right)$

$=\underset{x\to 0}{lim}\left(\frac{\sin x-x\cos x}{x\sin x}\right)$ ( $\frac{0}{0}$form)

using L'hospital rule

$=\underset{x\to 0}{lim}\frac{\cos x-\cos x+x\sin x}{\sin x+x\cos x}$

$=\underset{x\to 0}{lim}\frac{x\sin x}{\sin x+x\cos x}$ ($\frac{0}{0}$form)

again using L'Hospital rule

$=\underset{x\to 0}{lim}\frac{\sin x+x\cos x}{2\cos x-x\sin x}$

$=0$