Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

This question already has an answer here:

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

Thanks.

share|improve this question
3  
What have you tried? –  ᴊ ᴀ s ᴏ ɴ Jun 25 '13 at 9:08
2  
What is bigger: $3$ or $5$? Think about it. –  Cortizol Jun 25 '13 at 9:15
add comment

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

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

2 Answers

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}

share|improve this answer
add comment

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

share|improve this answer
add comment

Not the answer you're looking for? Browse other questions tagged or ask your own question.