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lim-x-0-1-cos-x-x-sin-x-




Question Number 193458 by SAMIRA last updated on 14/Jun/23
lim_(x→0)  (((1−cos(x))/(x sin(x)))) =   ???
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{1}−\mathrm{cos}\left(\mathrm{x}\right)}{\mathrm{x}\:\mathrm{sin}\left(\mathrm{x}\right)}\right)\:=\:\:\:??? \\ $$
Commented by aba last updated on 14/Jun/23
lim_(x→0) ((1−cos(x))/(xsin(x)))=lim_(x→0) ((1−cos(x))/x^2 )×(x/(sin(x)))=(1/2)×1=0.5
$$\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}−\mathrm{cos}\left(\mathrm{x}\right)}{\mathrm{xsin}\left(\mathrm{x}\right)}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}−\mathrm{cos}\left(\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} }×\frac{\mathrm{x}}{\mathrm{sin}\left(\mathrm{x}\right)}=\frac{\mathrm{1}}{\mathrm{2}}×\mathrm{1}=\mathrm{0}.\mathrm{5} \\ $$
Answered by MM42 last updated on 14/Jun/23
lim_(x→0)  (((1/2)x^2 )/x^2 ) =(1/2)
$${lim}_{{x}\rightarrow\mathrm{0}} \:\frac{\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{2}} }{{x}^{\mathrm{2}} }\:=\frac{\mathrm{1}}{\mathrm{2}} \\ $$

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