lim-x-0-x-1-x-2-sin-x-1-x-2-x-1-x-3-

Question Number 176035 by cortano1 last updated on 11/Sep/22

\lim_{x \to 0} \frac{x \sqrt{1 + x^{2}} - \sin x}{{(1 + x^{2})}^{x} - \sqrt{1 + x^{3}}} =

Answered by Ar Brandon last updated on 11/Sep/22

L = \lim_{x \to 0} \frac{x \sqrt{1 + x^{2}} - \sin x}{{(1 + x^{2})}^{x} - \sqrt{1 + x^{3}}} = \lim_{x \to 0} \frac{x (1 + \frac{x^{2}}{2}) - (x - \frac{x^{3}}{3!})}{(1 + x^{3}) - (1 + \frac{x^{3}}{2})}

= \lim_{x \to 0} \frac{x + \frac{x^{3}}{2} - (x - \frac{x^{3}}{6})}{\frac{x^{3}}{2}} = \lim_{x \to 0} \frac{\frac{x^{3}}{2} + \frac{x^{3}}{6}}{\frac{x^{3}}{2}} = \frac{(4 / 6)}{1 / 2} = \frac{4}{3}

Commented by cortano1 last updated on 11/Sep/22

Yes