calculus - Test $x_n=frac{n^2 e^{-sqrt{n}}}{cos(1/n)}$ for convergence and give its limit if possible.

Test $x_n=\frac{n^2 e^{-\sqrt{n}}}{\cos(1/n)}$ for convergence and give its limit if possible.

Easy part first: $1/n\rightarrow 0$ as $n\rightarrow\infty$ and hence $\cos(1/n)\rightarrow\cos(0)=1$.

Now the "harder" part. Would you say this arugment is sufficient/correct?

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We rewrite the numerator as $n^2 \frac{1}{e^\sqrt n}$. Since $e^\sqrt n$ grows faster than $n^2$ the term converges towards $0$ for $n\rightarrow\infty$.

So we have $lim_{n\rightarrow\infty}x_n=0/1=0$.