Question Number 197496 by sciencestudentW last updated on 19/Sep/23
$$\underset{{x}\rightarrow\mathrm{3}} {\mathrm{lim}}\frac{{cos}\mathrm{3}−{cosx}}{{x}−\mathrm{3}}=? \\ $$
Answered by cortano12 last updated on 20/Sep/23
$$\:=\:\underset{{x}\rightarrow\mathrm{3}} {\mathrm{lim}}\:\frac{−\mathrm{2sin}\:\left(\frac{\mathrm{x}+\mathrm{3}}{\mathrm{2}}\right)\:\mathrm{sin}\:\left(\frac{\mathrm{3}−\mathrm{x}}{\mathrm{2}}\right)}{−\mathrm{2}\left(\frac{\mathrm{3}−\mathrm{x}}{\mathrm{2}}\right)} \\ $$$$\:=\:\underset{{x}\rightarrow\mathrm{3}} {\mathrm{lim}}\:\mathrm{sin}\:\left(\frac{\mathrm{x}+\mathrm{3}}{\mathrm{2}}\right)\:.\:\underset{{x}\rightarrow\mathrm{3}} {\mathrm{lim}}\left(\frac{\mathrm{sin}\:\left(\frac{\mathrm{3}−\mathrm{x}}{\mathrm{2}}\right)}{\left(\frac{\mathrm{3}−\mathrm{x}}{\mathrm{2}}\right)}\right) \\ $$$$\:=\:\mathrm{sin}\:\mathrm{3}\:.\left(\mathrm{1}\right)=\:\mathrm{sin}\:\mathrm{3} \\ $$
Commented by MM42 last updated on 19/Sep/23
$${Sin}\mathrm{3}\:\vdots \\ $$
Answered by chengulapetrom last updated on 25/Sep/23
$$={lim}_{{x}\rightarrow\mathrm{3}} \frac{{sinx}}{\mathrm{1}} \\ $$$$={sin}\:\mathrm{3} \\ $$