[tex] \lim\limits_{x\to0}\dfrac{\sin x}{\sin\pi x}\Rightarrow\left(\dfrac{0}{0}\right)\\\\\text{Use the L'Hospital's rule}\ \lim\limits_{x\to c}\dfrac{f(x)}{g(x)}=\lim\limits_{x\to c}\dfrac{f'(x)}{g'(x)}\\\\(\sin x)'=\cos x\\\\(\sin\pi x)'=\cos\pi x\cdot\pi=\pi\cos\pi x\\\\\lim\limits_{x\to0}\dfrac{\sin x}{\sin\pi x}=\lim\limits_{x\to0}\dfrac{\cos x}{\pi\cos\pi x}=\dfrac{\cos0}{\pi\cos0}=\dfrac{1}{\pi\cdot1}=\dfrac{1}{\pi} [/tex]
