Question

$\underset{x\to 0}{\lim }$ $\frac{cosecx-cotx}{x}$ is

• A. $\frac{-1}{2}$
• B. $1$
• C. $\frac{1}{2}$
• D. $-1$

Answer 1

Answer: (C)

$\underset{x\to 0}{\lim }$ $\frac{cosecx-cotx}{x}$
$=\underset{x\to 0}{\lim }$ $\left\{\frac{\frac{1}{sinx}-\frac{cosx}{sinx}}{x}\right\}$
$=\underset{x\to 0}{\lim }$ $\left(\frac{1-cosx}{xsinx}\right)$
$=\underset{x\to 0}{\lim }$ $\frac{1-1+2si{n}^2\left(x/2\right)}{x\times 2sin\left(\frac{x}{2}\right)cos\left(\frac{x}{2}\right)}$
$=\underset{x\to 0}{\lim }$ $\frac{sin\left(\frac{x}{2}\right)}{xcos\left(\frac{x}{2}\right)}$
$=\underset{\frac{x}{2}\to 0}{\lim }$$\frac{tan\frac{x}{2}}{\frac{x}{2}}\times \frac{1}{2}$
$=\frac{1}{2}\times 1=1/2$