Let's $X \sim Exp(2)$ be a random variable. Compute $\lim_{n\to\infty} \frac{1}{E(X^{n})} $

Question

Probability density function of X is $f(x)=2e^{-2x}1_{(0,\infty)} (x)$. Then $E(X^{n})=\int_{0}^{\infty} x^n2e^{-2x} dx =...=\frac{1}{2^{n-2}}\cdot n!\cdot \frac{1}{4}.$ Because of that we have $\lim_{n\to\infty} \frac{1}{E(X^{n})} =0.$ I hope that solution is correct :)