TITLE: Can we solve this limit without squeeze theorem
QUESTION [5 upvotes]: I was wondering if we can solve this limit without using squeeze (sandwich) theorem.
$$\lim_{n\to \infty}(3^n+5^n)^{2/n}$$

REPLY [4 votes]: You may just write, as $n$ tends to $+\infty$,
$$
\begin{align}
(3^n+5^n)^{2/n}&=e^{\frac2n\log \left(3^n+5^n \right)}\\\\
&=e^{\frac2n\log \left(5^n\right)+ \frac2n\log \left(1+(3/5)^n \right)}\\\\
&=e^{2\log 5+ \frac2n \log \left(1+(3/5)^n \right)}\\\\
&=e^{2\log 5+ \frac2n (3/5)^n }\\\\
&\sim e^{2\log 5}\times e^0\\\\
&\sim25
\end{align}
$$ where we have used 
$$
\log (1+x)\sim_0 x
$$ with $x:=(3/5)^n$, $n$ being great.