Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

Question Number 182829 by yaslm last updated on 14/Dec/22

Answered by cortano1 last updated on 15/Dec/22

$$L=\underset{x\to \frac{\pi }{6}}{\lim }\frac{2\sin x-1}{\sqrt{3}\sec x-2}$$$$=\underset{x\to \frac{\pi }{6}}{\lim }\frac{\cos x(2\sin x-1)}{\sqrt{3}-2\cos x}$$$$=\frac{1}{2}\sqrt{3}\times \underset{x\to \frac{\pi }{6}}{\lim }\frac{2\sin x-1}{\sqrt{3}-2\cos x}$$$$[letx=\frac{\pi }{6}+y]$$$$L=\frac{1}{2}\sqrt{3}\underset{y\to 0}{\lim }\frac{2\sin (y+\frac{\pi }{6})-1}{\sqrt{3}-2\cos (y+\frac{\pi }{6})}$$$$=\frac{1}{2}\sqrt{3}\times \underset{y\to 0}{\lim }\frac{2(\frac{1}{2}\sqrt{3}\sin y+\frac{1}{2}\cos y)-1}{\sqrt{3}-2(\frac{1}{2}\sqrt{3}\cos y-\frac{1}{2}\sin y)}$$$$=\frac{1}{2}\sqrt{3}\times \underset{y\to 0}{\lim }\frac{\sqrt{3}\sin y+\cos y-1}{\sqrt{3}-\sqrt{3}\cos y+\sin y}$$$$=\frac{1}{2}\sqrt{3}\times \underset{y\to 0}{\lim }\frac{\sqrt{3}\sin y-2\sin {}^2\left(\frac{1}{2}y\right)}{2\sqrt{3}\sin {}^2\left(\frac{1}{2}y\right)+\sin y}$$$$=\frac{1}{2}\sqrt{3}\times \underset{y\to 0}{\lim }\frac{2\sqrt{3}\cos \left(\frac{1}{2}y\right)-2\sin \left(\frac{1}{2}y\right)}{2\sqrt{3}\sin \left(\frac{1}{2}y\right)+2\cos \left(\frac{1}{2}y\right)}$$$$=\frac{1}{2}\sqrt{3}\times \frac{2\sqrt{3}}{2}=\frac{3}{2}$$

Answered by ARUNG_Brandon_MBU last updated on 15/Dec/22

$$\mathcal{L}=\underset{x\to \frac{\pi }{6}}{\lim }\frac{2\sin x-1}{\sqrt{3}\sec x-2}=\underset{x\to \frac{\pi }{6}}{\lim }\frac{2\sin x\cos x-\cos x}{\sqrt{3}-2\cos x}$$$$=\underset{x\to \frac{\pi }{6}}{\lim }\frac{\sin 2x-\cos x}{\sqrt{3}-2\cos x}=\underset{t\to 0}{\lim }\frac{\sin (\frac{\pi }{3}-2t)-\cos (\frac{\pi }{6}-t)}{\sqrt{3}-2\cos (\frac{\pi }{6}-t)}$$$$=\underset{t\to 0}{\lim }\frac{\frac{\sqrt{3}}{2}\cos 2t-\frac{1}{2}\sin 2t-\frac{\sqrt{3}}{2}\cos t-\frac{1}{2}\sin t}{\sqrt{3}-\sqrt{3}\cos t-\sin t}$$$$=\underset{t\to 0}{\lim }\frac{\frac{\sqrt{3}}{2}(1-2t^2)-t-\frac{\sqrt{3}}{2}(1-\frac{t^2}{2})-\frac{t}{2}}{\sqrt{3}-\sqrt{3}(1-\frac{t^2}{2})-t}$$$$=\underset{t\to 0}{\lim }\frac{-\sqrt{3}{t}^2-\frac{3t}{2}+\frac{\sqrt{3}{t}^2}{4}}{\frac{t^2}{2}-t}=\frac{3}{2}$$

Answered by malwan last updated on 15/Dec/22

$$\underset{x\to \frac{\pi }{6}}{lim}\frac{2(sinx-\frac{1}{2})}{\frac{\sqrt{3}}{cosx}-2}$$$$=\underset{x\to \frac{\pi }{6}}{lim}\frac{2cosx(sinx-sin\frac{\pi }{6})}{2(cos\frac{\pi }{6}-cosx)}$$$$=\underset{x\to \frac{\pi }{6}}{lim}\frac{\frac{\sqrt{3}}{2}(2cos(\frac{x}{2}+\frac{\pi }{12})sin(\frac{x}{2}-\frac{\pi }{12})}{-2sin(\frac{\pi }{12}+x)sin(\frac{\pi }{12}-\frac{x}{12})}$$$$=\frac{\frac{\sqrt{3}}{2}\times \frac{\sqrt{3}}{2}}{\frac{1}{2}}=\frac{3}{2}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com