Interesting Limit (5)

Relevant wiki: Squeeze Theorem

Note that $\pi=\sqrt[n]{\pi^n}\le \sqrt[n]{\pi^n+e^n}\le \sqrt[n]{\pi^n+\pi^n}=\pi\sqrt[n]2$

And $\lim_{n\to\infty}\pi\sqrt[n]2=\pi$

By the squeeze theorem, we get $\lim_{n\to\infty}\sqrt[n]{\pi^n+e^n}=\boxed\pi$

Relevant wiki: L'Hopital's Rule - Basic

$\begin{aligned} L & = \lim_{n \to \infty} \sqrt[n]{\pi^n+e^n} \\ & = \lim_{n \to \infty} \exp \left(\ln (\sqrt[n]{\pi^n+e^n})\right) & \small \color{#3D99F6} \text{where }\exp (x) = e^x \\ & = \exp \left( \lim_{n \to \infty} \frac {\ln (\pi^n+e^n)}n \right) & \small \color{#3D99F6} \text{A }\infty/\infty \text{ case, L'Hôpital's rule applies.} \\ & = \exp \left( \lim_{n \to \infty} \frac {\frac {\pi^n \ln \pi+e^n}{\pi^n+e^n}}1 \right) & \small \color{#3D99F6} \text{Differentiate up and down w.r.t. }n. \\ & = \exp \left( \lim_{n \to \infty} \frac {\pi^n \ln \pi+e^n}{\pi^n+e^n} \right) & \small \color{#3D99F6} \text{Divide up and down by }\pi^n. \\ & = \exp \left( \lim_{n \to \infty} \frac {\ln \pi+\left(\frac e\pi\right)^n}{1+\left(\frac e\pi\right)^n} \right) \\ & = \exp (\ln \pi) \\ & = \boxed{\pi} \end{aligned}$