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Find the limit $$\mathop {\lim }\limits_{n \to \infty } \sqrt[n]{{{3^n} + {5^n}}}.$$


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marked as duplicate by mrf, Nick Peterson, Hagen von Eitzen, tetori, Thomas Andrews Jul 24 '13 at 6:29

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What is bigger: $3$ or $5$? Think about it. –  Cortizol Jun 25 '13 at 9:15

2 Answers 2

I think simple inequalities give a better intuition of what's going on,

\begin{align} 5=\sqrt[n]{5^n}\leq \sqrt[n]{3^n+5^n} \leq\sqrt[n]{5^n+5^n} = 5\sqrt[n]{2} \to 5 \end{align}

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We have $$ \lim_{n\rightarrow \infty}\sqrt[n]{{{3^n} + {5^n}}}=5\times\lim_{n\rightarrow \infty}\left[1+\left(\frac{3}{5}\right)^n\right]^{\frac{1}{n}}= 5.$$

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