Question

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

• A. 4
• B. e
• C. 5
• D. none of these.

Answer 1

Answer: (C)

${\lim }_{n\to \infty }{\left(4^n+5^n\right)}^{\frac{1}{n}}={\lim }_{n\to \infty }{\left(5^n\right)}^{\frac{1}{n}}{\left[1+{\left(\frac{4}{5}\right)}^n\right]}^{\frac{1}{n}}$
= $5{\lim }_{n\to \infty }{\left[{\left(1+{\left(\frac{4}{5}\right)}^n\right)}^{{\left(\frac{5}{4}\right)}^n}\right]}^{\frac{1}{n}{\left(\frac{4}{5}\right)}^n}$
= $5{\left(e\right)}^0=5\left(1\right)=5$
$\left[\because \frac{1}{n}\to 0asn\to \infty and{\left(\frac{4}{5}\right)}^n\to 0asn\to \infty \right]$