How can I calculate the limit of this? What is the limit? $$\lim_{n\rightarrow\infty}\dfrac{3}{(4^n+5^n)^{\frac{1}{n}}}$$
I don't get this limit. Really, I don't know if it has limit.

HINT: $$\begin{align*}
\frac3{(4^n+5^n)^{1/n}}&=\frac3{(4^n+5^n)^{1/n}}\cdot\frac{\left((1/5)^n\right)^{1/n}}{\left((1/5)^n\right)^{1/n}}\\\\
&=\frac3{5\left(1+\left(\frac45\right)^n\right)^{1/n}}
\end{align*}$$
Can you take the limit of that last expression as $n\to\infty$?
Here’s an intuitive way to think about it. When $n$ is very large, $4^n$ is a very small fraction of $5^n$, so $\left(4^n+5^n\right)^{1/n}$ ought to be just a little more than $\left(5^n\right)^{1/n}=5$.
