Evaluate $\lim_{n \to \infty} \sqrt[n]{3^n+4^n}$

$$\lim _{ n\rightarrow \infty }{ \sqrt [ n ]{ 3^{ n }+4^{ n } } } =4\cdot \lim _{ n\rightarrow \infty }{ \sqrt [ n ]{ \left( \frac { 3 }{ 4 } \right) ^{ n }+1 } } =4$$

Your proof is fine, provided you can use that $$ \lim_{n\to\infty}\sqrt[n]{2}=1 $$ This follows from Bernoulli’s inequality $(1+x)^n\ge 1+nx$, whenever $x>-1$ and $n$ is a positive integer, in the form $$ \sqrt[n]{1+nx}\le 1+x $$ For $x=1/n$ this reads $$ \sqrt[n]{2}\le 1+\frac{1}{n} $$ and therefore, from $$ 1\le\sqrt[n]{2}\le 1+\frac{1}{n} $$ and the squeeze theorem, you can conclude.

Then your application of the squeeze theorem to $$ 4=\sqrt[n]{4^n}\le\sqrt[n]{3^n+4^n}\le \sqrt[n]{2\cdot 4^n}=4\sqrt[n]{2} $$ is good.