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

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

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

$$\mathcal{L}=\underset{x\to 0}{\lim }\frac{\tan x}{\sqrt[3]{{(1-\cos x)}^2}}=\underset{x\to 0}{\lim }\frac{x+\frac{x^3}{3}}{\sqrt[3]{\frac{x^4}{4}}}$$$$=\sqrt[3]{4}\cdot \underset{x\to 0}{\lim }(x^{1-\frac{4}{3}}+\frac{1}{3}{x}^{3-\frac{4}{3}})$$$$=\sqrt[3]{4}\cdot \underset{x\to 0}{\lim }(\frac{1}{\sqrt[3]{x}}+\frac{\sqrt[3]{x^5}}{3})\to \pm \mathrm{\infty }$$

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