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Question Number 84377 by john santu last updated on 12/Mar/20
lim_(x→0)  (((√(1+tan x))−(√(1+sin x)))/(x^2 sin x))
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{1}+\mathrm{tan}\:\mathrm{x}}−\sqrt{\mathrm{1}+\mathrm{sin}\:\mathrm{x}}}{\mathrm{x}^{\mathrm{2}} \mathrm{sin}\:\mathrm{x}} \\ $$
Answered by john santu last updated on 12/Mar/20
lim_(x→0)  (1/( (√(1+tan x))+(√(1+sin x)))) × lim_(x→0)  ((tan x−sin x)/(x^2 sin x))  (1/2) × lim_(x→0)  (((1/2)x^3 +o(x^3 ))/x^3 ) = (1/4)
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\mathrm{tan}\:\mathrm{x}}+\sqrt{\mathrm{1}+\mathrm{sin}\:\mathrm{x}}}\:×\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{tan}\:\mathrm{x}−\mathrm{sin}\:\mathrm{x}}{\mathrm{x}^{\mathrm{2}} \mathrm{sin}\:\mathrm{x}} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\:×\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\frac{\mathrm{1}}{\mathrm{2}}\mathrm{x}^{\mathrm{3}} +\mathrm{o}\left(\mathrm{x}^{\mathrm{3}} \right)}{\mathrm{x}^{\mathrm{3}} }\:=\:\frac{\mathrm{1}}{\mathrm{4}} \\ $$

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