Differential and Integral Calculus by Feliciano and Uy, Exercise 1.2, Problem 4

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PROBLEM:

Evaluate lim⁡x→π3(sin⁡ 2xsin⁡ x)\displaystyle \lim\limits _{x\to \frac{\pi }{3}}\left(\frac{\sin\:2x}{\sin\:x}\right).

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SOLUTION:

Plug in the value x=π3\displaystyle x=\frac{\pi }{3}.

lim⁡x→π3(sin⁡ 2xsin⁡ x)=sin⁡(2⋅π3)sin⁡ (π3)=3232=1  (Answer)\begin{align*} \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)} \\ & =\frac{\frac{\sqrt{3}}{2}}{\frac{\sqrt{3}}{2}}\\ & =1 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \end{align*}
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