lim-x-0-tan-x-1-cos-x-2-1-3-

Question Number 161760 by cortano last updated on 22/Dec/21

\lim_{x \to 0} \frac{\tan x}{\sqrt[3]{{(1 - \cos x)}^{2}}} = ?

Answered by Ar Brandon last updated on 22/Dec/21

L = \lim_{x \to 0} \frac{\tan x}{\sqrt[3]{{(1 - \cos x)}^{2}}} = \lim_{x \to 0} \frac{x + \frac{x^{3}}{3}}{\sqrt[3]{\frac{x^{4}}{4}}}

= \sqrt[3]{4} \cdot \lim_{x \to 0} (x^{1 - \frac{4}{3}} + \frac{1}{3} x^{3 - \frac{4}{3}})

= \sqrt[3]{4} \cdot \lim_{x \to 0} (\frac{1}{\sqrt[3]{x}} + \frac{\sqrt[3]{x^{5}}}{3}) \to \pm \infty