Question

Evaluate $\underset{x\to 0}{lim}\left[\log \right(1+{\sin }^{2}x).\cot ({\log }^2(1+x)\left)\right]$

Answer 1

$\underset{x\to 0}{lim}\left[\log \right(1+{\sin }^{2}x).\cot ({\log }^2(1+x)\left)\right]$
$=\underset{x\to 0}{lim}\frac{\log (1+{\sin }^{2}x)}{\tan \left({\log }^2\right(1+x\left)\right)}=\underset{x\to 0}{lim}\frac{\frac{\log (1+{\sin }^{2}x)}{{\sin }^{2}x}.{\sin }^{2}x}{\frac{\tan \left({\log }^2\right(1+x\left)\right)}{{\log }^2(1+x)}.{\log }^2(1+x)}$
$=\underset{x\to 0}{lim}\frac{\frac{\log (1+{\sin }^{2}x)}{{\sin }^{2}x}.{\left(\frac{\sin x}{x}\right)}^2}{\frac{\tan \left({\log }^2\right(1+x\left)\right)}{{\log }^2(1+x)}.{\left(\frac{\log (1+x)}{x}\right)}^2}=\frac{1\times 1}{1\times 1}=1$