Limit of the sequence $\lim_{n\to\infty}{\sqrt[n]{4^n + 5^n}}$

Question

I have been trying to evaluate this limit:

$$\lim_{n\to\infty}{\sqrt[n]{4^n + 5^n}}$$

What methods should I try in order to proceed?

I was advised to use "Limit Chain Rule", but I believe there is a different approach.

Answer 1

Hint: $$\lim_{n\to\infty}{\sqrt[n]{4^n + 5^n}}=\lim_{n\to\infty}5{\sqrt[n]{\frac{4}{5^n}^n + 1}}$$

Answer 2

HINT:

Courtesy of the Sandwich theorem, $\displaystyle\lim_{n\to\infty}(a^n+b^n)^{1/n}=\max (a,b)\quad$ where $a,b>0$.

Answer 3

By the root test for a power series, this equivalent to the reciprocal of the radius of convergence of the following:

$$\sum_{n=0}^\infty(4^n+5^n)x^n$$

Which is a basic geometric series with radius of convergence $1/5$, so the limit is

$$\lim_{n\to\infty}\sqrt[n]{4^n+5^n}=1/R=5$$