Show convergence in distribution by the continuity theorem So the problem I'm about to solve is to show that:
$X \in \Gamma(a,b)$. Show that
\begin{equation}
\frac{X-E[X]}{\sqrt{Var(X)}} \xrightarrow{d} N(0,1)
\end{equation}
as $a \rightarrow \infty$, by using the continuity theorem.
I've made it this far:
\begin{equation}
\left(\frac{\exp{\left(-\frac{it}{\sqrt{a}}\right)}}{1-\frac{it}{\sqrt{a}}}\right)^{a}
\end{equation}
and now I want to show the limit when $a \rightarrow \infty$.
I have checked with a WolframAlpha, and the limit goes to $e^{-\frac{t^{2}}{2}}$, as expected, but I have no clue how to show this.

Hint: You could consider  
$$\exp(-x) = 1 - x + \dfrac{x^2}{2!} - \dfrac{x^3}{3!} +\cdots$$  
and 
$$(1+x)^k = 1 + kx + \dfrac{k(k-1)}{2!}x^2+ \cdots$$  
and 
$$\lim_{y \to \infty} \left(1-\frac{x}{y}\right)^y =\exp(-x)$$  
