<s>[INST] How can I evaluate $\lim_{v \to \frac{\pi}{3}}\frac{1-2\cos{v}}{\sin{\left(v-\frac{\pi}{3}\right)}}$ without using L'Hospital's rule? I found the following limit problem in Differential and Integral Calculus by Piskunov:
$$\lim\limits_{v \to \frac{\pi}{3}}\dfrac{1-2\cos{v}}{\sin{\left(v-\dfrac{\pi}{3}\right)}}$$
I tried searching for a trigonometric identity that could simply this, but couldn't make any work. How can this be solved without the application of L'Hospital's rule?
 [/INST] just use Taylor expansions:
$
\cos(x) = \frac{1}{2} - \frac{\sqrt{3} \cdot (x - \frac{\pi}{3})}{2} + o\left( x - \frac{\pi}{3}\right), \ x \to \frac{\pi}{3}
$
$
\sin(x) = x - \frac{\pi}{3}, \ o\left( x - \frac{\pi}{3}\right), \ x \to \frac{\pi}{3}
$
$
\lim\limits_{x \to \frac{\pi}{3}} \frac{1 - 2\cos(x)}{\sin(x - \frac{\pi}{3})} = \lim\limits_{x \to \frac{\pi}{3}} \frac{1 - 1 + \sqrt{3} \cdot (x - \frac{\pi}{3}) + o(x - \frac{\pi}{3})}{x - \frac{\pi}{3} + o(x - \frac{\pi}{3})} = \lim\limits_{x \to \frac{\pi}{3}} \sqrt{3} + o(1) = \boxed{\sqrt{3}}
$
</s>