Question

How do you solve this?

Answer 1

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