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

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

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?

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

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

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 $5$.

$$\lim_{n\to\infty}\sqrt[n]{3^n+5^n}=$$ $$\lim_{n\to\infty}\left(3^n+5^n\right)^{\frac{1}{n}}=$$ $$\lim_{n\to\infty}\exp\left(\ln\left(\left(3^n+5^n\right)^{\frac{1}{n}}\right)\right)=$$ $$\lim_{n\to\infty}\exp\left(\frac{\ln\left(3^n+5^n\right)}{n}\right)=$$ $$\exp\left(\lim_{n\to\infty}\frac{\ln\left(3^n+5^n\right)}{n}\right)=$$ $$\exp\left(\lim_{n\to\infty}\frac{\ln\left(5^n\right)+\ln\left(1+\left(\frac{3}{5}\right)^n\right)}{n}\right)=$$ $$\exp\left(\lim_{n\to\infty}\frac{\ln\left(5^n\right)}{n}\right)=$$ $$\exp\left(\lim_{n\to\infty}\frac{n\ln\left(5\right)}{n}\right)=$$ $$\exp\left(\lim_{n\to\infty}\frac{\ln\left(5\right)}{1}\right)=$$ $$\exp\left(\lim_{n\to\infty}\ln\left(5\right)\right)=$$ $$\exp\left(\ln\left(5\right)\right)=e^{\ln\left(5\right)}=5$$