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Question Number 176486 by cortano1 last updated on 20/Sep/22

$$\underset{x\to 0}{\lim }[\frac{\tan \mathrm{x}-\mathrm{x}}{{\mathrm{x}}^5}-\frac{1}{{3\mathrm{x}}^2}]=?$$

Answered by blackmamba last updated on 20/Sep/22

$$=\underset{x\to 0}{\lim }\frac{3(\tan x-x)-x^3}{3x^5}$$$$=\underset{x\to 0}{\lim }\frac{3((x+\frac{1}{3}{x}^3+\frac{2}{15}{x}^5+...)-x)-x^3}{3x^5}$$$$=\underset{x\to 0}{\lim }\frac{\left(\frac{2}{5}\right)x^5}{3x^5}=\frac{2}{15}$$

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