> [!NOTE] Theorem (Sum 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**: Let $(x_{n})_{n\geq 0}$ be a [[Real sequences|sequence of real numbers]] in $I\setminus\{ c \}$ such that $x_{n}\to c.$ Then by [[Sum Rule For Limits of Real Sequences]], $f(x_{n})+g(x_{n})\to \lim_{ n \to \infty}f(x_{n})+ \lim_{ n \to \infty } g(x_{n})= \lim_{ x \to c } f(x) + \lim_{ x \to c } g(x) $using [[Limit of Real Function by Convergent Real Sequences]]. Applying this again gives, $\lim_{ x \to c } (f(x)+g(x)) = \lim_{ x \to c } f(x) + \lim_{ x \to c } g(x)$