Showing a certain limit or an upper bound of it goes to zero

Question

I have the following limit which I know is equal to zero (using Mathematica soft.), however, I can't show it analytically.

$\lim_{L \rightarrow \infty} (\ln{L})^2\left[1-\left(1-e^{-\frac{1}{2}(\sqrt{L}-2)}\right)^L\right]$

I would appreciate any help. Thanks

Answer 1

For $L\ge4$, Bernoulli's Inequality says $$ \begin{align} \log(L)^2\left[1-\left(1-e^{-\frac12(\sqrt{L}-2)}\right)^L\right] &\le\log(L)^2\left[1-\left(1-Le^{-\frac12(\sqrt{L}-2)}\right)\right]\\ &=\log(L)^2Le^{-\frac12(\sqrt{L}-2)} \end{align} $$

Answer 2

Assume that $L$ is natural number

Hence $$\lim_{L \rightarrow \infty}\ (\ln{L})^2\left[1-\left(1-L e^{-\frac{1}{2}(\sqrt{L}-2)}\right)\right] =\lim_{M\rightarrow \infty}\ M^ka^{M}=0$$ for some $0<a<1$ and some $k>1$