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lim-x-0-2tan-x-5sin-x-3x-x-3-1-1-2x-3-1-5-




Question Number 124915 by liberty last updated on 06/Dec/20
 lim_(x→0)  ((2tan x−5sin x+3x−x^3 )/(1−((1−2x^3 ))^(1/5) )) =?
$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2tan}\:{x}−\mathrm{5sin}\:{x}+\mathrm{3}{x}−{x}^{\mathrm{3}} }{\mathrm{1}−\sqrt[{\mathrm{5}}]{\mathrm{1}−\mathrm{2}{x}^{\mathrm{3}} }}\:=?\: \\ $$
Answered by bemath last updated on 07/Dec/20
 lim_(x→0)  ((2(x+(x^3 /3))−5(x−(x^3 /6))+3x−x^3 )/(1−(1−((2x^3 )/5)))) =   lim_(x→0)  ((((2x^3 )/3) + ((5x^3 )/6)−x^3 )/((2x^3 )/5)) = lim_(x→0)  (((1/2)x^3 )/((2x^3 )/5)) = (5/4)
$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2}\left({x}+\frac{{x}^{\mathrm{3}} }{\mathrm{3}}\right)−\mathrm{5}\left({x}−\frac{{x}^{\mathrm{3}} }{\mathrm{6}}\right)+\mathrm{3}{x}−{x}^{\mathrm{3}} }{\mathrm{1}−\left(\mathrm{1}−\frac{\mathrm{2}{x}^{\mathrm{3}} }{\mathrm{5}}\right)}\:= \\ $$$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\frac{\mathrm{2}{x}^{\mathrm{3}} }{\mathrm{3}}\:+\:\frac{\mathrm{5}{x}^{\mathrm{3}} }{\mathrm{6}}−{x}^{\mathrm{3}} }{\frac{\mathrm{2}{x}^{\mathrm{3}} }{\mathrm{5}}}\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{3}} }{\frac{\mathrm{2}{x}^{\mathrm{3}} }{\mathrm{5}}}\:=\:\frac{\mathrm{5}}{\mathrm{4}} \\ $$

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