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.

Denote the function 
$$
f(n) = \frac{3}{(4^n +5^n)^{\frac{1}{n}}}
$$
Recall logarithm is a continuous function, hence denote 
$$
L(f(n)) = \log 3-\frac{\log(4^n +5^n)}{n}\\
\lim_{n \to \infty} L(f(n)) = \log 3 - \lim_{n \to \infty}\frac{\log(4^n +5^n)}{n}=\log 3 - \lim_{n \to \infty} \frac{4^n \log 4 + 5^n \log 5}{4^n + 5^n} \\
=\log 3 - \log 5=\log \bigg(\frac{3}{5} \bigg)
$$
I used here L'Hospital's rule and then divided the fraction through $5^n$. 
Hence $\lim_{n \to \infty}f(n)=\frac{3}{5}$
