Limit (Function)

analysis

Definition

Limit (Function)

Let $f:D\subseteq \mathbb{R}\to \mathbb{R}$ a function. The limit of $f$ is defined as:

$$\underset{x\to x_0}{\lim }f(x)=L:⟺\forall \epsilon >0:\exists \delta >0:\forall x\in D:|x-x_0|>\delta ⟹|f(x)-L|<\epsilon$$

Or equivalently:

$$\underset{x\to x_0}{\lim }f(x)=L⟺\forall (x_n{)}_{n\in \mathbb{N}}\subseteq D^{\mathbb{N}}:\underset{n\to \infty }{\lim }{x}_n=x_0⟹f(x_n)=L$$