Find the limit of ((1+3/(5n))/(1+4/(5n)))^(2+n/3) as n approaches infinity

 Exercise

$\lim_{n\to\infty}\left(\frac{1+\frac{3}{5n}}{1+\frac{4}{5n}}\right)^{2+\frac{n}{3}}$

 

Apply the power rule of limits: $\displaystyle{\lim_{x\to a}f(x)^{g(x)} = \lim_{x\to a}f(x)^{\displaystyle\lim_{x\to a}g(x)}}$

${\left(\lim_{n\to\infty }\left(\frac{1+\frac{3}{5n}}{1+\frac{4}{5n}}\right)\right)}^{\lim_{n\to\infty }\left(2+\frac{n}{3}\right)}$

 

Evaluate the limit $\lim_{n\to\infty }\left(2+\frac{n}{3}\right)$ by replacing all occurrences of $n$ by $\infty $

${\left(\lim_{n\to\infty }\left(\frac{1+\frac{3}{5n}}{1+\frac{4}{5n}}\right)\right)}^{\infty }$

 

Evaluate the limit $\lim_{n\to\infty }\left(\frac{1+\frac{3}{5n}}{1+\frac{4}{5n}}\right)$ by replacing all occurrences of $n$ by $\infty $

$1^{\infty }$

 

$1^\infty$ represents an indeterminate form

indeterminate

indeterminate

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