Question

Evaluate $\begin{array}{c}lim\\ n\to \infty \end{array}(4^n+5^n{)}^{1/n}$.

Answer 1

$\underset{n\to \infty }{lim}(4^n+5^n{)}^{1/n}$

$\Rightarrow \underset{n\to \infty }{lim}(4^n{)}^{1/n}{(1+{\left(\frac{5}{4}\right)}^n)}^{1/n}$

${\left(\frac{5}{4}\right)}^n>>>1$

$\Rightarrow \underset{n\to \infty }{lim}4{\left({\left(\frac{5}{4}\right)}^n\right)}^{1/n}$

$\Rightarrow \underset{n\to \infty }{lim}4\left(\frac{5}{4}\right)$

$\Rightarrow 5$.