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} \rightarrow 0 \, as\, n \rightarrow \infty \, and \, \left(\frac{4}{5}\right)^{n} \rightarrow 0 \, as \, n \rightarrow \infty\right] $