Show that $\lim_{n\to\infty} \sqrt[n]{a^{n} + b^{n}} = \max(a, b)$

Question

Prove that:

$$\lim_{n\to\infty} \sqrt[n]{a^{n} + b^{n}} = \max(a, b)$$

I don't have a clue even how to start the proof here. Any hint is appreciated!

Answer 1

Assume $a\geq b\geq0$ and $a>0$ (if $a=b=0$ the statement is trivial). Write $$ \lim_{n\to\infty} \sqrt[n]{a^{n} + b^{n}} = \lim_{n\to\infty}\left( a\cdot\sqrt[n]{1 + (b/a)^{n}} \right), $$ with $1\geq (b/a)\geq (b/a)^n$. Then observe that $$ 1\leq\sqrt[n]{1 + (b/a)^{n}}\leq\sqrt[n]{2}\leq 1+1/n, $$ and conclude that the limit is exactly $a$.

Answer 2

Hint: Whenever $x>y>0$, $x^n>>y^n$, so the larger term dominates inside the radical, and you can "ignore" the smaller term for large enough $n$ and evaluate easily.

Answer 3

Without loss of generality, assume $a \geq b \geq 0$, then we have the following inequality: $$\sqrt[n]{a^n} \leq \sqrt[n]{a^n + b^n} \leq \sqrt[n]{a^n + a^n}.$$ Now apply the squeeze principle and the fact $\lim_{n \to \infty} 2^{1/n} = 1$.