Mathematics - Limits Question with Solution | TestHub

$\begin{array}{rl}& l=\underset{x\to 0^{+}}{\lim }(\sin (\sin kx)+\cos x+x{)}^{\frac{2}{x}}=e^6\\ & \Rightarrow \ln l=\underset{x\to 0^{+}}{\lim }\frac{2}{x}(\sin (\sin kx)+\cos x+x-1)\\ & \Rightarrow \ln l=\underset{x\to 0^{+}}{\lim }2\left(\frac{\sin (\sin kx)}{\sin kx}\cdot \frac{\sin kx}{kx}\cdot \frac{kx}{x}+1-\frac{(1-\cos x)}{x^2}\cdot x\right)\\ & \Rightarrow \ln l=2(k+1)\\ & \Rightarrow l=e^{2(k+1)}=e^6\\ & k+1=3\\ & \Rightarrow k=2\end{array}$