<s>[INST] Strategy for the Limit: $\lim_{n\to\infty} \frac{2^{n+1}+3^{n+1}}{2^n+3^n} $ I do not understand how to properly solve this limit: 
$$
\lim_{n\to\infty} \frac{2^{n+1}+3^{n+1}}{2^n+3^n}
$$
I thought of breaking it up: 
$$
\lim_{n\to\infty} \frac{2^{n+1}}{2^n+3^n} +\lim_{n\to\infty} \frac{3^{n+1}}{2^n+3^n}
$$
But I do not see how this will allow me to use any of the limit rules to reduce. I know that the series converges though. 
Thanks. 
 [/INST] Since the 3s dominate,
I would expect the limit
to be 3.
Check:
$\begin{array}\\
\dfrac{2^{n+1}+3^{n+1}}{2^n+3^n}-3
&=\dfrac{2^{n+1}+3^{n+1}-3(2^n+3^n)}{2^n+3^n}\\
&=\dfrac{2^{n+1}+3^{n+1}-3\cdot 2^n-3^{n+1}}{2^n+3^n}\\
&=\dfrac{2^{n+1}-3\cdot 2^n}{2^n+3^n}\\
&=\dfrac{-2^n}{2^n+3^n}\\
&\to 0
\qquad\text{since } 2^n/3^n \to 0\\
\end{array}
$
</s>