Evaluate $\lim\limits_{x \to \frac{\pi}{2}} \; \left(1+\cos x\right)^{3 \sec x}$
Let $\cos x = t$
Then as $x \to \dfrac{\pi}{2}, \; t \to 0$
$\begin{aligned}
\lim\limits_{x \to \frac{\pi}{2}} \; \left(1+\cos x\right)^{3 \sec x} & = \lim\limits_{x \to \frac{\pi}{2}} \; \left(1+\cos x\right)^{\frac{3}{\cos x}} \\
& = \lim\limits_{t \to 0} \; \left(1+t\right)^{\frac{3}{t}} \\
& = \left\{\lim\limits_{t \to 0} \; \left(1+t\right)^{\frac{1}{t}}\right\}^3 \\
& = e^3 \;\; \left[\text{Note: }\lim\limits_{x \to 0} \left(1+x\right)^{\frac{1}{x}}=e\right]
\end{aligned}$