Question Number 126744 by bemath last updated on 24/Dec/20
$$\:\:{Given}\:{f}\left({x}\right)=\:\frac{\mathrm{2}−\sqrt{\mathrm{3}}\:\mathrm{cos}\:{x}−\mathrm{sin}\:{x}}{\left(\mathrm{6}{x}−\pi\right)^{\mathrm{2}} } \\ $$$$\:{Find}\:\underset{{x}\rightarrow\frac{\pi}{\mathrm{6}}} {\mathrm{lim}}\:{f}\left({x}\right)\:. \\ $$
Answered by liberty last updated on 24/Dec/20
$$\:\underset{{x}\rightarrow\pi/\mathrm{6}} {\mathrm{lim}}\frac{\sqrt{\mathrm{3}}\:\mathrm{sin}\:{x}−\mathrm{cos}\:{x}}{\mathrm{12}\left(\mathrm{6}{x}−\pi\right)}\:=\:\underset{{x}\rightarrow\pi/\mathrm{6}} {\mathrm{lim}}\frac{\sqrt{\mathrm{3}}\:\mathrm{cos}\:{x}+\mathrm{sin}\:{x}}{\mathrm{72}} \\ $$$$\:=\:\frac{\sqrt{\mathrm{3}}\:\mathrm{cos}\:\left(\frac{\pi}{\mathrm{6}}\right)+\mathrm{sin}\:\left(\frac{\pi}{\mathrm{6}}\right)}{\mathrm{72}}\:=\:\frac{\frac{\mathrm{3}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}}{\mathrm{72}}\:=\:\frac{\mathrm{1}}{\mathrm{36}}\: \\ $$
Answered by bemath last updated on 24/Dec/20
