What is the limit of $ \sqrt[n]{ a^ n + b^ n}$ when $n \to \infty$ for $a,b ≥ 0$

Question

I want to caclulate $$\lim_{n\rightarrow \infty} {\sqrt[n]{ a^ n + b^ n } }$$ for a,b ≥ 0

I wanted to isolate $\sqrt[n]{a^n}$ but I really don't know how to show that the limit is 1.

I know that $\sqrt[n]a \rightarrow 1$ ,so I thought that I seperate the function to $\sqrt[n]{a^n}$ and $\sqrt[n]{b^n}$ and show that they are equally going to the same limit. So that $\sqrt{a^n}$ and $\sqrt{b^n} = \lim{\sqrt[n]{ a^{ n } + b^{ n } } }$.

Answer 1

The limit is $\max(a,b)$. Assume WLOG that $a> b$. Then

$$\lim_{n\rightarrow \infty} {\sqrt[n]{ a^ n + b^ n } }=\lim_{n\rightarrow \infty}a{\sqrt[n]{ 1 + \left(\frac{b}{a}\right)^n } }=a$$

since

$$\lim_{n\rightarrow \infty}{\sqrt[n]{ 1 + \left(\frac{b}{a}\right)^n } }=1$$

On the other hand, if $a=b$ then

$$\lim_{n\rightarrow \infty} {\sqrt[n]{ a^ n + b^ n } }=\lim_{n\rightarrow \infty} {\sqrt[n]{2a^n} }=\lim_{n\rightarrow \infty}2^{1/n}a=a$$