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  4. Evaluate displaystyle lim n arrow ∞(4n+5n)(1/n)

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Q. Evaluate $\displaystyle\lim _{n \rightarrow \infty}\left(4^{n}+5^{n}\right)^{\frac{1}{n}}$

Limits and Derivatives

Solution:

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