Showing \(f(x)\) = sin \(x\) Is Continuous at 0
\(f(0) = \sin 0 = 0\), so \(f\) is defined at 0. \(\lim\limits_{x\rightarrow 0}f(x) = \lim\limits_{x\rightarrow 0}\sin x = 0\), so the limit at 0 exists. \(\lim\limits_{x\rightarrow 0}\sin x = \sin 0 = 0\).
Since all three conditions of continuity are satisfied, \(f(x) = \sin x\) is continuous at 0.