Question

Solve :
$\underset{x\to 0}{lim}(1+{\tan }^2\sqrt{x}{)}^{1/2x}$

Answer 1

$\begin{array}{c}{lim}_{x\to 0}{\left(1+{\tan }^2\sqrt{x}\right)}^{\frac{1}{2x}}=e^{{lim}_{x\to 0}\frac{\tan \sqrt[2]{2x}}{2x}}\\ =e^{{lim}_{x\to 0}\frac{\tan \sqrt[2]{2x}}{{\left(\sqrt{x}\right)}^2}\times \frac{1}{2}}=e\frac{1}{2}\end{array}$

Hence,

solved.