Question

The value of $\underset{n\to \infty }{lim}{\left(4^n+5^n\right)}^{1/n}$ is equal to

• A. $5$
• B. $4$
• C. $0$
• D. D.N.E.

Answer 1

The correct option is A $5$

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

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

Using indetermined form of $1^{\infty }$ ,we get

$L=\underset{n\to \infty }{lim}5e^{\frac{1}{n}\times (1+{\left(\frac{4}{5}\right)}^n-1)}=\underset{n\to \infty }{lim}5e^{{\left(\frac{4}{5}\right)}^n\times \frac{1}{n}}$

Using L-Hospital, we get

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