Mathematics - Limits Question with Solution | TestHub

Here, ${\lim }_{x\to 0}(\sin x{)}^{1/x}+{\lim }_{x\to 0}{\left(\frac{1}{x}\right)}^{\sin x}\Rightarrow 0+{\lim }_{x\to 0}{e}^{\log {\left(\frac{1}{x}\right)}^{\sin x}}\left\{\begin{array}{l}{\lim }_{x\to 0}(\sin x{)}^{1/x}\to 0\\ \text{ as, (decimal) }{)}^x\to 0\end{array}\right\}$ $\Rightarrow e^{{\lim }_{x\to 0}\frac{\log (1/x)}{\mathrm{cosec}x}}$ On applying L Hospital's Rule, we get $$\Rightarrow e^{{\lim }_{x\to 0}\frac{x\left(-\frac{1}{x^2}\right)}{-\mathrm{cosec}x\cot x}}=e^{{\lim }_{x\to 0}\frac{\sin x}{x}\tan x}=e^0=1$$