Differential and Integral Calculus by Feliciano and Uy: Limit of a Function, Exercise 1.2, Problem 4

Evaluate \lim\limits _{x\to \frac{\pi }{3}}\left(\frac{sin\:2x}{sin\:x}\right).

SOLUTION:

Plug in the value x=\frac{\pi }{3}

\lim\limits_{x\to \frac{\pi }{3}}\left(\frac{sin\:2x}{sin\:x}\right)=\frac{sin\left(2\cdot \frac{\pi }{3}\right)}{sin\:\left(\frac{\pi }{3}\right)}

\lim\limits_{x\to \frac{\pi }{3}}\left(\frac{sin\:2x}{sin\:x}\right)=\frac{\frac{\sqrt{3}}{2}}{\frac{\sqrt{3}}{2}}

\lim\limits_{x\to \frac{\pi }{3}}\left(\frac{sin\:2x}{sin\:x}\right)=1

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