lim-x-1-x-x-2-1-x-x-2-

Question Number 128605 by john_santu last updated on 08/Jan/21

\lim_{x \to - \infty} \sqrt{1 + x + x^{2}} - \sqrt{1 - x + x^{2}} = ?

Commented by john_santu last updated on 08/Jan/21

Answered by mathmax by abdo last updated on 09/Jan/21

f (x) = \sqrt{x^{2} + x + 1} - \sqrt{x^{2} - x + 1} \Rightarrow

f (x) =∣ x ∣ \sqrt{1 + \frac{1}{x} + \frac{1}{x^{2}}} - ∣ x ∣ \sqrt{1 - \frac{1}{x} + \frac{1}{x^{2}}}

for x \to - \infty f (x) \sim - x (1 + \frac{1}{2} (\frac{1}{x} + \frac{1}{x^{2}})) + x (1 + \frac{1}{2} (- \frac{1}{x} + \frac{1}{x^{2}})

= - \frac{1}{2} - \frac{1}{2 x} - \frac{1}{2} + \frac{1}{2 x} = - 1 \Rightarrow \lim_{x \to - \infty} f (x) = - 1