L'Hôpital's Rule states that the [[Limits|Limit]] of any [[Function]] over another is equal to their [[Derivative|derivative]] divided over each other. $ \huge\lim_{x\to a} \frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)} $ >[!example] >$\huge\begin{align*} \lim_{x\to1}\frac{\sin(\pi x)}{x^{2}-1} &= \lim_{x\to1} \frac{ \frac{\mathrm{d}(\sin(\pi x))}{\mathrm{d}x} }{ \frac{\mathrm{d}(x^{2}-1)}{\mathrm{d}x} } \\ &= \lim_{x\to1} \frac{\pi\cos(\pi x)}{2x} \\ &= \frac{\pi\cos(\pi)}{2} \\ &= -\frac{\pi}{2} \end{align*}$