a nice limit : part 2

Relevant wiki: L'Hopital's Rule - Basic

$\begin{aligned} L & = \lim_{n \to \infty}(1+n^2)^{\frac 1{\ln(n)}} \\ & = \lim_{n \to \infty} \exp \left(\ln \left((1+n^2)^{\frac 1{\ln(n)}}\right) \right) & \small \color{#3D99F6} \text{where }\exp (x) = e^x \\ & = \exp \left(\lim_{n \to \infty} \frac {\ln (1+n^2)}{\ln(n)} \right) & \small \color{#3D99F6} \text{A }\infty/\infty \text{ case, L'Hôpital's rule applies.} \\ & = \exp \left(\lim_{n \to \infty} \frac {\frac {2n}{1+n^2}}{\frac 1n} \right) & \small \color{#3D99F6} \text{Differentiate up and down w.r.t. }n. \\ & = \exp \left(\lim_{n \to \infty} \frac {2n^2}{1+n^2} \right) & \small \color{#3D99F6} \text{Divide up and down by }n^2. \\ & = \exp \left(\lim_{n \to \infty} \frac 2{\frac 1{n^2}+1} \right) \\ & = \boxed{e^2} \end{aligned}$