> [!NOTE] Theorem (Product Rule) > Let $I$ be an [[Open Real Interval|open real interval]]. Let $c\in I.$ Let $f$ be a [[Real Function|real function]] defined on $I,$ except possibly at $c.$ Let the [[Limit of Real Function at a Point|limits]] of $f(x)$ and $g(x)$ as $x$ tends to $c$ exist. Then $\lim_{ x \to c }(f(x)g(x))=\lim_{ x \to c }f(x)\lim_{ x \to c }g(x).$ **Proof**: