Question

Lim +infinito x(√x^2-1-x)

Answer 1

$\lim_{x \to \infty} \frac{x}{\sqrt{x^2-1}-x} \\\\ \text{multiplica pelo conjugado do denominador}\\\\ \lim_{x \to \infty} \frac{x}{\sqrt{x^2-1}-x} * \frac{\sqrt{x^2-1}+x}{\sqrt{x^2-1}+x} \\\\ \lim_{x \to \infty} \frac{x*(\sqrt{x^2-1}+x)}{(\sqrt{x^2-1}-x)(\sqrt{x^2-1}+x)} \\\\ \to\text{diferenca dos quadrados no denominador} (a-b)(a+b)=a^2-b^2\\\\ \lim_{x \to \infty} \frac{x*(\sqrt{x^2-1}+x)}{(\sqrt{x^2-1})^2-x^2} \\\\ \lim_{x \to \infty} \frac{x*(\sqrt{x^2-1}+x)}{(x^2-1)-x^2}$

$\lim_{x \to \infty} \frac{x*(\sqrt{x^2-1}+x)}{-1}\\\\ \lim_{x \to \infty} - x(\sqrt{x^2-1}+x) = -\infty$