Find the limit of (3+2/x)^(4x) as x approaches infinity

 Exercise

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

 

Rewrite the limit using the identity: $a^x=e^{x\ln\left(a\right)}$

$\lim_{x\to\infty }\left(e^{4x\ln\left(3+\frac{2}{x}\right)}\right)$

 

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

${\left(\lim_{x\to\infty }\left(e\right)\right)}^{\lim_{x\to\infty }\left(4x\ln\left(3+\frac{2}{x}\right)\right)}$

 

The limit of a constant is just the constant

$e^{\lim_{x\to\infty }\left(4x\ln\left(3+\frac{2}{x}\right)\right)}$

 

Evaluate the limit $\lim_{x\to\infty }\left(4x\ln\left(3+\frac{2}{x}\right)\right)$ by replacing all occurrences of $x$ by $\infty $

$e^{\infty }$

 

Apply a property of infinity: $k^{\infty}=\infty$ if $k>1$. In this case $k$ has the value $e$

$\infty $

$\infty $

 Try other ways to solve this exercise

Can't find a method? Tell us so we can add it.