> [!NOTE] Definition (Convergence of real function at a point) > Let $I \subset \mathbb{R}$ 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 $L\in \mathbb{R}.$ Then $f(x)$ tends to $L$ as $x$ tends to $c,$ denoted $\lim_{ x \to c } f(x) = L,$iff for all $\varepsilon>0,$ there exists $\delta>0$ such that for all $x\in I,$ $0<|x-c|<\delta \implies |f(x)-L|<\varepsilon.$ > > [!Example] Example (Why do we want $|x-c|>0$?) > Consider $f(x)=\begin{cases} x^{2} & x \neq 0 \\ 1 & x= 0 \end{cases}$Do we want $\lim_{ x \to 0 }f(x)$ to exist? If you want it to exist and be zero, you need to rule out $x=c$ in the definition. # Properties See [[Limit of Real Function by Convergent Real Sequences]] which relates limit of a function at a point to [[Convergence]] in its domain at infinity. Note that a function is [[Continuous Function (Epsilon-Delta Definition)|continuous]] iff $\lim_{ x \to c } f(x) = f(c)$. Algebra of function limits is the same as [[Algebra of Limits of Convergent Sequences]]. # Applications - The [[Fréchet Differentiation]] of a real-valued function of a single real variable and [[Fréchet Differentiation]] of real-valued function of multiple real-variables. - See [[Asymptotic Notations]].