How to show $\lim_{n \to \infty} \sqrt[n]{a^n+b^n}=\max \{a,b\}$? [duplicate]

real-analysis sequences-and-series limits radicals means

Hint: $$\max\{a,b\} \leq \sqrt[n]{a^n+b^n} \leq \sqrt[n]{2\max\{a,b\}^n} = 2^{1/n}\max\{a,b\}$$ and make $n \to +\infty.$ Here's a bit more general version of this result.


Assume without loss of generality $a > b$. Then you get

$$\sqrt[n]{a^n+b^n} = \sqrt[n]{a^n\cdot(1 + \frac{b^n}{a^n})} = a\cdot \sqrt[n]{(1 + \frac{b^n}{a^n})},$$

now conclude the limit using that $(\frac{b}{a})^n$ tends to $0$ for $n\to \infty$.

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