Definition:Limit

Sequences
Let $$\left \langle {x_n} \right \rangle$$ be a sequence in $\mathbb{R}$.

Let $$\left \langle {x_n} \right \rangle$$ converge to a value $$l \in \mathbb{R}$$.

Then $$l$$ is known as the limit of $$\left \langle {x_n} \right \rangle$$ as $$n$$ tends to infinity and is usually written:

$$l = \lim_{n \to \infty} x_n$$

Functions
Let $$f$$ be a real function defined on an open interval including $$c$$ (except possibly at $$c$$). Let $$L$$ be a real number.

This means that $$\forall \epsilon \in \mathbb{R}^+, \exists \delta \in \mathbb{R}^+$$ such that $$0 < |x-c| < \delta \Longrightarrow |f(x)-L| < \epsilon$$.

$$\lim_{x \to c} f \left({x}\right)$$ is voiced "the limit of $$f \left({x}\right)$$ as $$x$$ tends to $$c$$".