lim-x-x-2-x-2-4x-x-3-

Question Number 99951 by bemath last updated on 24/Jun/20

\lim_{x \to - \infty} x^{2} \sqrt{x^{2} + 4 x} + x^{3} ?

Commented by bobhans last updated on 24/Jun/20

\lim_{x \to - \infty} \sqrt{x^{6} + {4 x}^{5}} + x^{3} \times {\frac{\sqrt{x^{6} + {4 x}^{5}} - x^{3}}{\sqrt{x^{6} + {4 x}^{5}} - x^{3}}} =

\lim_{x \to - \infty} \frac{{4 x}^{5}}{\sqrt{x^{6} + {4 x}^{5}} - x^{3}} = \lim_{x \to - \infty} \frac{- 4}{\sqrt{\frac{1}{x^{4}} + \frac{4}{x^{5}}} + \frac{1}{x^{2}}} = - \infty

Answered by mathmax by abdo last updated on 24/Jun/20

f (x) = x^{2} \sqrt{x^{2} + 4 x} + x^{3} \Rightarrow for x < 0 we get f (x) = x^{2} ∣ x ∣ \sqrt{1 + \frac{4}{x}} + x^{3}

= - x^{3} \sqrt{1 + \frac{4}{x}} + x^{3} \sim - x^{3} (1 + \frac{2}{x}) + x^{3} = - {2 x}^{2} (x \to - \infty) \Rightarrow

\lim_{x \to - \infty} f (x) = - \infty