## Problem Statement

Calculate the limit of the numerical sequence:

$\lim _{n \rightarrow \infty} \frac{2 n-\sin n}{\sqrt{n}-\sqrt[3]{n^{3}-7}}$
## Solution

$$
\lim _{n \rightarrow \infty} \frac{2 n-\sin n}{\sqrt{n}-\sqrt[3]{n^{3}-7}}=\lim _{n \rightarrow \infty} \frac{\frac{1}{n}(2 n-\sin n)}{\frac{1}{n}\left(\sqrt{n}-\sqrt[3]{n^{3}-7}\right)}=
$$

$=\lim _{n \rightarrow x} \frac{2-\frac{\sin n}{n}}{\sqrt{\frac{1}{n}}-\sqrt[3]{1-\frac{7}{n^{3}}}}=$

Since $\sin n$ is bounded, then

$\sin n$

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

Then:

$=\lim _{n \rightarrow \infty} \frac{2-0}{\sqrt{0}-\sqrt[3]{1-0}}=\lim _{n \rightarrow \infty} \frac{2}{0-1}=-2$