## Limits of sequences

### Task number: 2870

Determine the following limits, or prove that they do not exist.

#### Variant 1

\(\displaystyle \lim_{n\to\infty} \frac{2n^2+4n+n \sin n}{ n \cos 3n + (2n +\sin n)^2}\)

#### Variant 2

\(\displaystyle \lim_{n\to\infty} \frac{\lfloor \sqrt{n} \rfloor}{\sqrt{n}}\)

#### Variant 3

\(\displaystyle \lim_{n\to\infty} \cos (n^2 \pi) +\cos ((n+1) \pi) \)

#### Variant 4

\(\displaystyle \lim_{n\to\infty} n^{\cos (n \pi)} \)

#### Variant 5

\(\displaystyle \lim_{n\to\infty} \frac{\sqrt[4]{n^5+2}-\sqrt[3]{n^2+1}}{\sqrt[5]{n^4+2}-\sqrt{n^3+1}}\)

#### Variant 6

\(\displaystyle \lim_{n\to\infty} \frac{3^n+n^5}{n^6+n!}\)

#### Variant 7

\(\displaystyle \lim_{n\to\infty} \frac{\sqrt[3]{n^2}\sin(n!)}{n+1}\)