Question

$\lim_{x \to +\infty} \ xe^{-x}$

$\lim_{x \to \infty} \frac{ln x}{x}$

Answer 1

Vamos aplicar a regra de Cauchy:

1 -

$\displaystyle \lim_{x \to +\infty} x \cdot e^{-x} \\ \\ \\ \lim_{x \to +\infty} x \cdot \frac{1}{e^{x}} \\ \\ \\ \lim_{x \to +\infty} \frac{x}{e^{x}} \\ \\ \\ \lim_{x \to +\infty} \frac{1}{e^{x}} \\ \\ \\ \lim_{x \to +\infty} \frac{1}{e^{\infty}} = 0$

2 - 

$\displaystyle \lim_{x \to +\infty} \frac{\ln(x)}{x} \\ \\ \\ \lim_{x \to +\infty} \frac{\displaystyle \frac{1}{x}}{1} \\ \\ \\ \lim_{x \to +\infty} \frac{1}{x} \\ \\ \\ \lim_{x \to +\infty} \frac{1}{\infty} = 0$