<s>[INST] $1 - \lim_{x \to \infty} \left( 1 - \frac{2}{x^2} \left( 1 - \exp \left( -\frac{x^2}{2} \right) \right) p \right)^{kx^2}$ I'm currently working on my master thesis and I need to solve this limit.
I forgot almost everything about limits since last time a I saw them was basically in high school.
Anyway, I solved this limit with wolfram and it gives:
$1 - \lim_{x \to \infty} \left( 1 - \frac{2}{x^2} \left( 1 - \exp \left(  -\frac{x^2}{2} \right) \right) p \right)^{kx^2} = 1 - \exp(-2 k p)$
$k$ and $p$ are two parameters and, in particular, $k>0$, $p \in (0,1)$.
I'd like to have a step-by-step solution of the limit, or at least I'd like to know which method I could use to solve it.
Thanks!
 [/INST] We shall use the expansion $\log(1+x) = x + O(x^2)$ as $x\rightarrow 0$.
Write $$
\begin{align*}\left( 1 - \frac{2}{x^2} \left( 1 - \exp \left(  -\frac{x^2}{2} \right) \right) p \right)^{kx^2} &= \exp\left\{kx^2\log \left(1-\frac{2p}{x^2}+\frac{2p}{x^2e^{x^2/2}}\right)\right\} \\
&= \exp\left(kx^2\left(-\frac{2p}{x^2}+\frac{2p}{x^2e^{x^2/2}} + O(x^{-4}) \right)\right) \text{ as } x \rightarrow \infty \\
&= \exp\left(-2pk + 2pke^{-x^2/2} + O(x^{-2})\right)\\
&\rightarrow \exp(-2pk) \text{ as } x \rightarrow \infty
\end{align*}
$$
Hence,
$$1-\lim_{x\rightarrow\infty}\left( 1 - \frac{2}{x^2} \left( 1 - \exp \left(  -\frac{x^2}{2} \right) \right) p \right)^{kx^2} = 1 - \exp(-2kp)$$
</s>