<s>[INST] Evaluating $\lim_{x\to0} \frac{\cos x - \cos 3x}{\sin 3x^2 - \sin x^2}$ 
$$ \lim_{x\to0}\frac{\cos x-\cos (3x)}{\sin (3x^2)-\sin (x^2)} $$

Is there a simple way of finding the limit? 
I know the long one: rewrite it as 
$$ -\lim_{x\to 0}\frac{\cos x-\cos(3x)}{\sin(3x^2)}\cdot\frac{1}{1-\dfrac{\sin(3x^2)}{\sin(x^2)}} $$
and then find both limits in separately applying L'Hospital's rule several times. The answer is $2$.
 [/INST] Hint: Use the factorisation formula $$\cos a -\cos b =-2\sin{a+b\over 2}\sin{a-b\over 2}$$
and
$$\sin a -\sin b =2\sin{a-b\over 2}\cos{a+b\over 2}$$
$$ \lim_{x\to0}\frac{\cos x-\cos (3x)}{\sin (3x^2)-\sin (x^2)} = \lim_{x\to0}\frac{2\color{red}{\sin 2x}\cdot \color{green}{\sin x}\cdot\color{blue}{x^2}}{\color{blue}{\sin (x^2)}\cos (2x^2)\cdot\color{red}{2x}\cdot \color{green}{x}} =2$$
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