> [!NOTE] Theorem (Algebra of Limits) > 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 > > (1) [[Sum Rule for Limits of Real Functions]]: $\lim_{ x \to c }(f(x)+g(x))=\lim_{ x \to c }f(x)+ \lim_{ x \to c }g(x).$ > > (2) [[Product Rule for Limits of Real Functions]]: $\lim_{ x \to c }(f(x)g(x))=\lim_{ x \to c }f(x)\lim_{ x \to c }g(x).$ > > (3) [[Quotient Rule for Limits of Real Functions]]: If $\lim_{ x \to c }g(x)\neq 0$ then $\lim_{ x \to c } \frac{f(x)}{g(x)}=\frac{\lim_{ x \to c }f(x)}{\lim_{ x \to c }g(x)}.$ **Proof**: All proofs make use of [[Limit of Real Function by Convergent Real Sequences]] and [[Algebra of Limits of Convergent Sequences]] .