## Problem Statement

Calculate the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty} \frac{2^{n+1}+3^{n+1}}{2^{n}+3^{n}}
$$
## Solution

$$
\begin{aligned}
& \lim _{n \rightarrow \infty} \frac{2^{n+1}+3^{n+1}}{2^{n}+3^{n}}=\lim _{n \rightarrow \infty} \frac{2 \cdot 2^{n}+3 \cdot 3^{n}}{2^{n}+3^{n}}= \\
& =\lim _{n \rightarrow \infty} \frac{2 \cdot 2^{n}+2 \cdot 3^{n}+3^{n}}{2^{n}+3^{n}}=\lim _{n \rightarrow \infty} \frac{2\left(2^{n}+3^{n}\right)+3^{n}}{2^{n}+3^{n}}= \\
& =\lim _{n \rightarrow \infty}\left(2+\frac{3^{n}}{2^{n}+3^{n}}\right)=2+\lim _{n \rightarrow \infty} \frac{\frac{1}{3^{n}} 3^{n}}{\frac{1}{3^{n}}\left(2^{n}+3^{n}\right)}= \\
& =2+\lim _{n \rightarrow \infty} \frac{1}{\left(\frac{2}{3}\right)^{n}+1}=2+\frac{1}{0+1}=2+1=3
\end{aligned}
$$

## Problem Kuznetsov Limits 6-4