Problem with limit $\lim_{n\rightarrow\infty} |\lambda_{i}|^{n} n^{\ell}$ How I can calculate next:
$$
\lim_{n\rightarrow\infty} |\lambda_{i}|^{n} n^{\ell} ? 
$$
Where $|\lambda_{i}|<1$ and $\ell$ is any positive integer.

Note that
$$
\lim_{n\to \infty} |\lambda_{i}|^{n} n^{\ell} = 
\left(\lim_{n\to \infty} |\lambda_{i}|^{n/\ell} n\right)^{\ell} = 
\left(\lim_{n\to \infty}  \frac{n}{e^{-[\ln(|\lambda_{i}|)/\ell]n}}\right)^{\ell}
$$
From there, L'Hôpital's rule, perhaps, noting $\ln(|\lambda_i|) < 0$.
