Question Number 182346 by cortano1 last updated on 08/Dec/22
Commented by SANOGO last updated on 08/Dec/22
$${thank}\:{you} \\ $$
Answered by CrispyXYZ last updated on 08/Dec/22
$$=\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\frac{\frac{\mathrm{3}\pi}{{x}^{\mathrm{2}} }\:\mathrm{sin}\left(\frac{\mathrm{3}\pi}{{x}}\right)}{\frac{\pi}{\mathrm{2}}\:\mathrm{cos}\left(\frac{\pi{x}}{\mathrm{2}}\right)} \\ $$$$=\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\frac{−\frac{\mathrm{3}\pi}{{x}^{\mathrm{4}} }\left(\mathrm{2}{x}\:\mathrm{sin}\left(\frac{\mathrm{3}\pi}{{x}}\right)+\mathrm{3}\pi\:\mathrm{cos}\left(\frac{\mathrm{3}\pi}{{x}}\right)\right)}{−\frac{\pi^{\mathrm{2}} }{\mathrm{4}}\:\mathrm{sin}\left(\frac{\pi{x}}{\mathrm{2}}\right)} \\ $$$$=\frac{−\mathrm{3}\pi\:\centerdot\:\left(−\mathrm{3}\pi\right)}{−\frac{\pi^{\mathrm{2}} }{\mathrm{4}}} \\ $$$$=−\mathrm{36} \\ $$