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