Question

Não entendi a explicação do professor
$\lim_{x \to \infty} (1+ \frac{3}{x})^{4x}$

Answer 1

 Apliquemos o Limite Fundamental Exponencial. A saber, $\lim_{x\to\infty}\left(1+\frac{1}{x}\right)^x=\lim_{y\to0}(1+y)^{\frac{1}{y}}=e$
 
 Desenvolvendo o teu limite,

- Consideremos $4x=\lambda\Leftrightarrow\,x=\frac{\lambda}{4}$, segue,

$\lim_{x\to\infty}\left(1+\frac{3}{x}\right)^{4x}=\\\\\\\lim_{\lambda\to\infty}\left(1+\frac{3}{\frac{\lambda}{4}}\right)^{\lambda}=\\\\\\\lim_{\lambda\to\infty}\left(1+\frac{12}{\lambda}\right)^{\lambda}=$
 
- Consideremos agora $\frac{12}{\lambda}=\mu\Leftrightarrow\lambda=\frac{12}{\mu}$, segue que,

$\lim_{\mu\to\0}\left(1+\mu\right)^{\frac{12}{\mu}}=\\\\\\\left[\lim_{\mu\to0}(1+\mu)^{\frac{1}{\mu}}\right]^{12}=\\\\\\\boxed{\boxed{e^{12}}}$