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How might i calculate the following limit

$\lim_{n\to\infty}\left(a^{n}+b^{n}\right)^{\frac{1}{n}}$

for positive and real numbers a,b. I belive the answer is max(a,b) from limit intuition however i dont know how to formally prove it. I couldnt figure out the algebra required to prove it through the epsilon definition and i couldnt find an expression to bound it from above with to use the squeeze theorom.

bookim online
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    When you have an intuition about a limit, use it! Either subtract from the expression, and show it tends to zero, or divide, and show it tends to one.

    In this case, it'll be easier to divide by $\max(a,b)$, and show this tends to 1.

     – Robbie Dec 01 '21 at 21:28

1 Answers1

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WLOG assume that $a > b$. Then

$$ (a^n + b^n)^{1/n} = a\cdot(1 + (b/a)^n)^{1/n}$$

where $0<b/a \leq 1$. But in general for $0\leq x\leq 1$ we have

$$ (1 + x^n)^{1/n} = \exp\left(\log\left(1 + x^n\right)/n\right) \rightarrow \exp(0) = 1 $$

as $n \rightarrow \infty$.

Bernard
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Kevin S. Van Horn
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