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lim-x-0-x-sin-x-x-5-2-




Question Number 84954 by john santu last updated on 17/Mar/20
lim_(x→0)  (((√x) − (√(sin x)))/x^(5/2) ) = ?
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{x}}\:−\:\sqrt{\mathrm{sin}\:\mathrm{x}}}{\mathrm{x}^{\frac{\mathrm{5}}{\mathrm{2}}} }\:=\:? \\ $$
Answered by john santu last updated on 17/Mar/20
lim_(x→0)  ((x−sin x)/(x^2 (√x))) ×(1/( (√x) + (√(sin x)))) =   lim_(x→0)  ((x−(x−(1/6)x^3 ))/(x^3  (1+(√((sin x)/x))))) =   lim_(x→0)  (((1/6)x^3 )/(x^3  (1+(√((sin x)/x))))) = ((1/6)/(1+1)) = (1/(12))
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{x}−\mathrm{sin}\:\mathrm{x}}{\mathrm{x}^{\mathrm{2}} \sqrt{\mathrm{x}}}\:×\frac{\mathrm{1}}{\:\sqrt{\mathrm{x}}\:+\:\sqrt{\mathrm{sin}\:\mathrm{x}}}\:=\: \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{x}−\left(\mathrm{x}−\frac{\mathrm{1}}{\mathrm{6}}\mathrm{x}^{\mathrm{3}} \right)}{\mathrm{x}^{\mathrm{3}} \:\left(\mathrm{1}+\sqrt{\frac{\mathrm{sin}\:\mathrm{x}}{\mathrm{x}}}\right)}\:=\: \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\frac{\mathrm{1}}{\mathrm{6}}\mathrm{x}^{\mathrm{3}} }{\mathrm{x}^{\mathrm{3}} \:\left(\mathrm{1}+\sqrt{\frac{\mathrm{sin}\:\mathrm{x}}{\mathrm{x}}}\right)}\:=\:\frac{\frac{\mathrm{1}}{\mathrm{6}}}{\mathrm{1}+\mathrm{1}}\:=\:\frac{\mathrm{1}}{\mathrm{12}} \\ $$
Commented by jagoll last updated on 18/Mar/20
waw... great
$$\mathrm{waw}…\:\mathrm{great} \\ $$

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