Solving a square sequence limit

Question

I've started learning sequences and I'm having a hard time calculating the following:

$$\lim_{n\to ∞}{\sqrt[n]{3^n + 7^n}}$$

Using Heine’s Lemma I'm trying to solve it analogous to the corresponding limit definitions for functions, but I get stuck every direction I go.

Any help is appreciated.

Answer 1

Hint: Factor $7^n$ out of the sum underneath the radical. Then simplify (you may find the fact that $0\leq\frac37\leq1$ useful (if you prefer to avoid l'Hopital).

Answer 2

$\lim_{n\to\infty}(3^n+7^n)^\frac1n=\lim_{n\to\infty}7(1+(\frac 37)^n)^\frac1n=7\cdot 1^0=7$