Question

$\displaystyle \lim_{x \to 0}$ $\frac{cosec\,x-cot\,x}{x}$ is

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

Answer 1

Answer: (C)

$\displaystyle \lim_{x \to 0}$ $\frac{cosec\,x-cot\,x}{x}$
$=\displaystyle \lim_{x \to 0}$ $\left\{\frac{\frac{1}{sin\,x}-\frac{cos\,x}{sin\,x}}{x} \right\}$
$=\displaystyle \lim_{x \to 0}$ $\left(\frac{1-cos\,x}{x\,sin\,x}\right)$
$=\displaystyle \lim_{x \to 0}$ $\frac{1-1+2\,sin^{2}\left(x/2\right)}{x \times 2\,sin \left(\frac{x}{2}\right)cos \left(\frac{x}{2}\right)}$
$=\displaystyle \lim_{x \to 0}$ $\frac{sin\left(\frac{x}{2}\right)}{x\,cos\left(\frac{x}{2}\right)}$
$=\displaystyle \lim_{\frac{x}{2} \to 0}$$\frac{tan \frac{x}{2}}{\frac{x}{2}}\times\frac{1}{2}$
$=\frac{1}{2}\times1=1/2$