Question and Answers Forum

All Questions      Topic List

Limits Questions

Previous in All Question      Next in All Question      

Previous in Limits      Next in Limits      

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 )

$$\mathrm{cos}\:^{\mathrm{2}} \left(\frac{\mathrm{3}{x}}{\mathrm{2}}\right)\:=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{cos}\:\mathrm{3}{x}+\mathrm{1}\right) \\ $$$$\underset{{x}\rightarrow\pi/\mathrm{6}} {\mathrm{lim}}\:\frac{\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{cos}\:\mathrm{3}{x}+\mathrm{1}\right)−\mathrm{sin}\:{x}}{\mathrm{sin}\:{x}+\sqrt{\mathrm{3}}\:\mathrm{cos}\:{x}−\mathrm{2}} \\ $$$$\underset{{x}\rightarrow\pi/\mathrm{6}} {\mathrm{lim}}\frac{\frac{\mathrm{1}}{\mathrm{2}}\left(−\mathrm{3sin}\:\mathrm{3}{x}\right)−\mathrm{cos}\:{x}}{\mathrm{cos}\:{x}−\sqrt{\mathrm{3}}\:\mathrm{sin}\:{x}} \\ $$$$=\frac{\frac{\mathrm{3}}{\mathrm{2}}+\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}{\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}−\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}\:=\:\infty\:\left({DNE}\:\right) \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com