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 184796 by Spillover last updated on 11/Jan/23

$$Evaluate$$$$\underset{x\to \frac{\pi }{6}}{\lim }\frac{\sqrt{3}\sin x-\cos x}{x-\frac{\pi }{6}}$$

Commented by MJS_new last updated on 11/Jan/23

$$2$$

Commented by MJS_new last updated on 11/Jan/23

$$\mathrm{these}\mathrm{are}\mathrm{all}\mathrm{easy}\mathrm{with}{\mathrm{l}}^{\prime }\mathrm{H}\widehat{\mathrm{o}pital}$$

Answered by aba last updated on 12/Jan/23

$$\mathrm{let}\mathrm{t}=\mathrm{x}-\frac{\pi }{6}\Rightarrow \mathrm{x}=\mathrm{t}+\frac{\pi }{6}$$$$\underset{x\to \frac{\pi }{6}}{\lim }\frac{\sqrt{3}\sin \left(\mathrm{x}\right)-\cos \left(\mathrm{x}\right)}{\mathrm{x}-\frac{\pi }{6}}=\underset{\mathrm{t}\to 0}{\lim }\frac{\sqrt{3}\sin (\mathrm{t}+\frac{\pi }{6})-\cos (\mathrm{t}+\frac{\pi }{6})}{\mathrm{t}}$$$$=\underset{\mathrm{t}\to 0}{\lim }\frac{\sqrt{3}(\sin \left(\mathrm{t}\right)\cos \left(\frac{\pi }{6}\right)+\cos \left(\mathrm{t}\right)\sin \left(\frac{\pi }{6}\right))-(\cos \left(\mathrm{t}\right)\cos \left(\frac{\pi }{6}\right)-\sin \left(\mathrm{t}\right)\sin \left(\frac{\pi }{6}\right))}{\mathrm{t}}$$$$=\underset{\mathrm{t}\to 0}{\lim }\frac{\sqrt{3}(\sqrt{3}\sin \left(\mathrm{t}\right)+\cos \left(\mathrm{t}\right))-(\sqrt{3}\cos \left(\mathrm{t}\right)-\sin \left(\mathrm{t}\right))}{2\mathrm{t}}$$$$=\underset{\mathrm{t}\to 0}{\lim }\frac{3\sin \left(\mathrm{t}\right)+\sin \left(\mathrm{t}\right)+\sqrt{3}\sin \left(\mathrm{t}\right)-\sqrt{3}\cos \left(\mathrm{t}\right)}{2\mathrm{t}}$$$$=\underset{\mathrm{t}\to 0}{\lim }\frac{4\sin \left(\mathrm{t}\right)}{2\mathrm{t}}$$$$=2$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com