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$\lim_{n\to \infty}\sqrt[n]{3^n+5^n}$

I'm stuck. Tried to apply the squeeze theorem, but found only the $\le$ side, which is $\sqrt[n]{5^n}$, approaching 5.

How do I proceed from there?

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marked as duplicate by Arjang, Guy Fsone, muaddib, M. Vinay, Chris Godsil Jan 28 '18 at 14:08

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.

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Hint: $$\sqrt[n]{3^n+5^n}=5\sqrt[n]{\left(\frac{3}{5}\right)^n+1}$$

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The answer is in the comment of David Mitra. $5^n\le 3^n+5^n\le 2*5^n$.

If you take limits, you get $5\le \lim\sqrt[n]{3^n+5^n}\le \lim\sqrt[n]{2}*5$. Since $\lim\sqrt[n]{2}=1$, you get that your limit is $5$.

Please upvote David Mitra's comment, not my answer.

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  • $\begingroup$ +1 for the attitude. I was writing a comment to all David to post his comment as an answer when you already posted it. But so glad about the acknowledgement $\endgroup$ – Shailesh Nov 9 '15 at 17:00
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$$\lim_{n\to\infty}\sqrt[n]{3^n+5^n}=$$ $$\lim_{n\to\infty}\left(3^n+5^n\right)^{\frac{1}{n}}=$$ $$\lim_{n\to\infty}\exp\left(\ln\left(\left(3^n+5^n\right)^{\frac{1}{n}}\right)\right)=$$ $$\lim_{n\to\infty}\exp\left(\frac{\ln\left(3^n+5^n\right)}{n}\right)=$$ $$\exp\left(\lim_{n\to\infty}\frac{\ln\left(3^n+5^n\right)}{n}\right)=$$ $$\exp\left(\lim_{n\to\infty}\frac{\ln\left(5^n\right)+\ln\left(1+\left(\frac{3}{5}\right)^n\right)}{n}\right)=$$ $$\exp\left(\lim_{n\to\infty}\frac{\ln\left(5^n\right)}{n}\right)=$$ $$\exp\left(\lim_{n\to\infty}\frac{n\ln\left(5\right)}{n}\right)=$$ $$\exp\left(\lim_{n\to\infty}\frac{\ln\left(5\right)}{1}\right)=$$ $$\exp\left(\lim_{n\to\infty}\ln\left(5\right)\right)=$$ $$\exp\left(\ln\left(5\right)\right)=e^{\ln\left(5\right)}=5$$

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  • $\begingroup$ Thank you! I wondered if there was a exp ln solution :) $\endgroup$ – ntrch Nov 10 '15 at 12:51

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