Question

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

Answer 1

Answer: 5

$\underset{n\to \infty }{\lim }{\left(4^n+5^n\right)}^{\frac{1}{n}}$
$=\underset{n\to \infty }{\lim }{\left[5^n{\left(1+\frac{4}{5}\right)}^n\right]}^{\frac{1}{n}}$
$=5\underset{n\to \infty }{\lim }{\left[1+{\left(\frac{4}{5}\right)}^n\right]}^{\frac{1}{n}}$
$=5(1)=5$