<s>[INST] How do I solve this limit without l'Hopital? I tried the substitution $t=x-(\pi/3)$ but it doesn't help at all. I have also tried using $\sin(\pi/3)=\sqrt{3}/2$ but couldn't do anything useful then. I tried to factor the denominator and numerator, but it didn't help either. I want a solution without l'Hopital's rule.
$$\lim_{x\to \pi/3} \left[\dfrac{\sin^2(x) - \sin^2\left(\dfrac{\pi}{3}\right)}{x^2 -\left(\dfrac{\pi}{3}\right)^2}\right]$$
 [/INST] We can rewrite the limit in the following way:
$$\lim_{x \to \frac{\pi}{3}} \frac{\sin^2{x} - \sin^2{\frac{\pi}{3}}}{x^2 - \left(\frac{\pi}{3}\right)^2} = 
\frac{2\sin{\frac{\pi}{3}}}{2\frac{\pi}{3}} lim_{x \to \frac{\pi}{3}} \frac{\sin{x} - \sin{\frac{\pi}{3}}}{x - \frac{\pi}{3}}=
\frac{\sin{\frac{\pi}{3}}}{\frac{\pi}{3}} lim_{x \to \frac{\pi}{3}} \frac{2\sin{(x-\frac{\pi}{3})}\cos{(x+\frac{\pi}{3})}}{x - \frac{\pi}{3}} = 
\frac{\sin{\frac{\pi}{3}}}{\frac{\pi}{3}}2 \cos{\frac{2\pi}{3}}$$
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