Menu Close

lim-x-pi-6-cos-2-3x-2-sin-x-sin-x-3-cos-x-2-




Question Number 105157 by bramlex last updated on 26/Jul/20
lim_(x→(π/6))  ((cos^2 (((3x)/2))−sin x)/(sin x+(√3) cos x−2)) ?
$$\underset{{x}\rightarrow\frac{\pi}{\mathrm{6}}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:^{\mathrm{2}} \left(\frac{\mathrm{3}{x}}{\mathrm{2}}\right)−\mathrm{sin}\:{x}}{\mathrm{sin}\:{x}+\sqrt{\mathrm{3}}\:\mathrm{cos}\:{x}−\mathrm{2}}\:? \\ $$
Answered by bramlex last updated on 26/Jul/20
cos^2 (((3x)/2)) = (1/2)(cos 3x+1)  lim_(x→π/6)  (((1/2)(cos 3x+1)−sin x)/(sin x+(√3) cos x−2))  lim_(x→π/6) (((1/2)(−3sin 3x)−cos x)/(cos x−(√3) sin x))  =(((3/2)+((√3)/2))/(((√3)/2)−((√3)/2))) = ∞ (DNE )
$$\mathrm{cos}\:^{\mathrm{2}} \left(\frac{\mathrm{3}{x}}{\mathrm{2}}\right)\:=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{cos}\:\mathrm{3}{x}+\mathrm{1}\right) \\ $$$$\underset{{x}\rightarrow\pi/\mathrm{6}} {\mathrm{lim}}\:\frac{\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{cos}\:\mathrm{3}{x}+\mathrm{1}\right)−\mathrm{sin}\:{x}}{\mathrm{sin}\:{x}+\sqrt{\mathrm{3}}\:\mathrm{cos}\:{x}−\mathrm{2}} \\ $$$$\underset{{x}\rightarrow\pi/\mathrm{6}} {\mathrm{lim}}\frac{\frac{\mathrm{1}}{\mathrm{2}}\left(−\mathrm{3sin}\:\mathrm{3}{x}\right)−\mathrm{cos}\:{x}}{\mathrm{cos}\:{x}−\sqrt{\mathrm{3}}\:\mathrm{sin}\:{x}} \\ $$$$=\frac{\frac{\mathrm{3}}{\mathrm{2}}+\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}{\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}−\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}\:=\:\infty\:\left({DNE}\:\right) \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *