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  • limit of a function: <img alt="\mathrm{\lim _{x \rightarrow \frac{\pi}{3}} \frac{\sin \left(x-\frac{\pi}{3}\right)}{1-2 \cos x}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5

limit of a function: \mathrm{\lim _{x \rightarrow \frac{\pi}{3}} \frac{\sin \left(x-\frac{\pi}{3}\right)}{1-2 \cos x}}

Option: 1

\frac{\sqrt3}{3}


Option: 2

\frac{\sqrt3}{2}


Option: 3

0


Option: 4

1


Answers (1)

best_answer

Let \mathrm{y=x-\pi / 3}. Then, the limit of interest becomes

\mathrm{\lim _{y \rightarrow 0} \frac{\sin y}{1-2 \cos (y+\pi / 3)} =\lim _{y \rightarrow 0} \frac{\sin y}{1-\cos y+\sqrt{3} \sin y} }

                                                      \mathrm{ =\lim _{y \rightarrow 0} \frac{2 \sin (y / 2) \cos (y / 2)}{2 \sin ^2(y / 2)+2 \sqrt{3} \sin (y / 2) \cos (y / 2)} }

\mathrm{ =\lim _{y \rightarrow 0} \frac{\cos (y / 2)}{\sin (y / 2)+\sqrt{3} \cos (y / 2)} }

\mathrm{ =\frac{\sqrt{3}}{3} }
Hence option 1 is correct.




 

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Pankaj

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