Let $X_1,...., X_n$ be a sequence of i.i.d random variables with p.d.f
$$ f(x)= \begin{cases} 4x^2 e^{-2x},& \text{if } x>0\\ 0, & \text{otherwise} \\ \end{cases}$$
and let $S_n=\sum _{i=1} ^n X_i$. Then $\lim_{n \rightarrow \infty} P(S_n \leq \frac{3n}{2}+\sqrt3n)$
How to proceed?
I assume it is $\sqrt{3n}$ instead of $\sqrt{3}n$ in your question. We have $$ P\left(S_n \leq \frac{3n}{2} + \sqrt{3n}\right) = P\left(\frac{S_n}{n} \leq \frac{3}{2} + \sqrt{\frac{3}{n}}\right) = P\left(\sqrt{n}\cdot\left(\frac{S_n}{n} - \frac{3}{2}\right) \leq \sqrt{3}\right) $$ Since $\mathsf{E}[X_i] = \frac{3}{2}$ and $\mathsf{Var}[X_i] = \frac{3}{4}$, by central limit theorem, the random variable $\sqrt{n}\cdot \left(\frac{S_n}{n} - \frac{3}{2}\right)$ follows the normal distribution $\mathcal{N}(0, 3/4)$ when $n$ goes to infinity. Then $$ \lim_{n\rightarrow\infty}P\left(S_n \leq \frac{3n}{2} + \sqrt{3n}\right) = \Phi(2) $$