Question

Evaluate the following limits:

${lim}_{x\to \pi }\frac{1+cosx}{ta{n}^{2}x}$

Answer 1

${lim}_{x\to \pi }\frac{1+cosx}{ta{n}^{2}x}$

$={lim}_{x\to \pi }[\frac{(1+cosx)}{si{n}^{2}x}\times co{s}^{2}x]$

$={lim}_{x\to \pi }[\frac{1+cosx}{1-co{s}^{2}x}\times co{s}^{2}x]$

$={lim}_{x\to \pi }[\frac{1+cosx}{(1-cosx)(1+cosx)}\times co{s}^{2}x]$

$={lim}_{x\to \pi }\left[\frac{co{s}^{2}x}{(1-cosx)}\right]$

$={lim}_{x\to \pi }\left[\frac{co{s}^2\pi }{1-cos\pi }\right]$

$=\frac{(-1{)}^2}{1-(-1)}=\frac{1}{2}$