## Problem Statement

Calculate the limit of the numerical sequence:

$\lim _{n \rightarrow \infty} \frac{\sqrt{n^{2}+3 n-1}+\sqrt[3]{2 n^{2}+1}}{n+2 \sin n}$
## Solution

$\lim _{n \rightarrow \infty} \frac{\sqrt{n^{2}+3 n-1}+\sqrt[3]{2 n^{2}+1}}{n+2 \sin n}=\lim _{n \rightarrow \infty} \frac{\frac{1}{n}\left(\sqrt{n^{2}+3 n-1}+\sqrt[3]{2 n^{2}+1}\right)}{\frac{1}{n}(n+2 \sin n)}=$
$=\lim _{n \rightarrow \infty} \frac{\sqrt{1+\frac{3}{n}-\frac{1}{n^{2}}}+\sqrt[3]{\frac{2}{n}+\frac{1}{n^{3}}}}{1+2 \frac{\sin n}{n}}=$

Since $\sin n$ is bounded, then

$\frac{\sin n}{n} \rightarrow 0$, as $n \rightarrow \infty$

Then:

$=\frac{\sqrt{1+0-0}+\sqrt[3]{0+0}}{1+2 \cdot 0}=\frac{\sqrt{1}+\sqrt[3]{0}}{1}=1$