Question Number 176486 by cortano1 last updated on 20/Sep/22
![lim_(x→0) [ ((tan x−x)/x^5 ) −(1/(3x^2 )) ]=?](https://www.tinkutara.com/question/Q176486.png)
$$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\left[\:\frac{\mathrm{tan}\:\mathrm{x}−\mathrm{x}}{\mathrm{x}^{\mathrm{5}} }\:−\frac{\mathrm{1}}{\mathrm{3x}^{\mathrm{2}} }\:\right]=? \\ $$
Answered by blackmamba last updated on 20/Sep/22
$$=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{3}\left(\mathrm{tan}\:{x}−{x}\right)−{x}^{\mathrm{3}} }{\mathrm{3}{x}^{\mathrm{5}} } \\ $$$$=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{3}\left(\left({x}+\frac{\mathrm{1}}{\mathrm{3}}{x}^{\mathrm{3}} +\frac{\mathrm{2}}{\mathrm{15}}{x}^{\mathrm{5}} +…\right)−{x}\right)−{x}^{\mathrm{3}} }{\mathrm{3}{x}^{\mathrm{5}} } \\ $$$$=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\frac{\mathrm{2}}{\mathrm{5}}\right){x}^{\mathrm{5}} }{\mathrm{3}{x}^{\mathrm{5}} }\:=\:\frac{\mathrm{2}}{\mathrm{15}} \\ $$