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Question Number 139001 by bramlexs22 last updated on 21/Apr/21

lim_(x→0) ((((1+sin^2 x))^(1/3) −((1−2tan x))^(1/4) )/(sin x+tan^2 x)) =?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt[{\mathrm{3}}]{\mathrm{1}+\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}}−\sqrt[{\mathrm{4}}]{\mathrm{1}−\mathrm{2tan}\:\mathrm{x}}}{\mathrm{sin}\:\mathrm{x}+\mathrm{tan}\:^{\mathrm{2}} \mathrm{x}}\:=? \\ $$

Answered by EDWIN88 last updated on 21/Apr/21

lim_(x→0) cos^2 x(((((1+sin^2 x))^(1/3) −((1−2tan x))^(1/4) )/(cos^2 x sin x +sin^2 x)))= lim_(x→0) (((1+(x^2 /3))−(1−((2x)/4)))/(x(1+x)))= lim_(x→0) ((x((x/3)+(1/2)))/(x(1+x))) = (1/2)

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{cos}^{\mathrm{2}} \:\mathrm{x}\left(\frac{\sqrt[{\mathrm{3}}]{\mathrm{1}+\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}}−\sqrt[{\mathrm{4}}]{\mathrm{1}−\mathrm{2tan}\:\mathrm{x}}}{\mathrm{cos}^{\mathrm{2}} \:\mathrm{x}\:\mathrm{sin}\:\mathrm{x}\:+\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}}\right)= \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{1}+\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{3}}\right)−\left(\mathrm{1}−\frac{\mathrm{2x}}{\mathrm{4}}\right)}{\mathrm{x}\left(\mathrm{1}+\mathrm{x}\right)}= \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{x}\left(\frac{\mathrm{x}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{2}}\right)}{\mathrm{x}\left(\mathrm{1}+\mathrm{x}\right)}\:=\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$

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