Question

Calcule usando Euler
lim   n--> ∞
 
(1 +4/n)^{n}

Answer 1

temos:
$\boxed{\boxed{ \lim_{n \to \infty} \left(1+ \frac{4}{n} \right)^n}}$

fazendo a substituição:
$u = \frac{4}{n} \to \text{quando n tende a infinito u tende a 0}\\\\ n= \frac{4}{u} \\\\ \boxed{\boxed{\lim_{n \to \infty} \left(1+ \frac{4}{n} \right)^n = \lim_{u \to 0} \left(1+ u \right)^ { \frac{4}{u} } }}$

observando o limite fundamental de euler:
$\boxed{\boxed{ \lim_{n \to \infty} \left(1+ \frac{1}{n} \right)^n=e}}$

fazendo a substituição nesse limite
$u = \frac{1}{n} \\\\ n = \frac{1}{u} \\\\\text{logo}\\\\\\ \boxed{\boxed{ \lim_{n \to \infty} \left(1+ \frac{1}{n} \right)^n= \lim_{n \to 0} \left(1+u \right)^ \frac{1}{n} =e}}$

sabendo disso podemos resolver a questão
$\lim_{n \to \infty} \left(1+ \frac{1}{n} \right)^n=\lim_{u \to 0} \left(1+ u \right)^ { \frac{4}{u}} = \left( \lim_{u \to 0} \left(1+ u \right)^ { \frac{1}{u} \right)^4 = (e)^4 = e^4$