Question

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

Answer 1

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

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

$\therefore L=\underset{n\to \infty }{lim}5{(\frac{4^n}{5^n}+1)}^{\frac{1}{n}}$

Now $\underset{n\to \infty }{lim}\frac{4^n}{5^n}=\underset{n\to \infty }{lim}{\left(\frac{4}{5}\right)}^n=0$

$\therefore L=\underset{n\to \infty }{lim}5(0+1{)}^{\frac{1}{n}}$

$\therefore L=\underset{n\to \infty }{lim}5(1{)}^{\circ }$

$\therefore L=5$

$\therefore$ The value of limit is zero (0)