Limit (mathematics)

The concept of the limit is a cornerstone of the mathematical branch of analysis and captures the intuitive concept of "approaching a value".

Limit of a Sequence

Let $(a_{n})_{n\in N}=(a_{1},a_{2},a_{3},...)$ be a sequence of real numbers. We say that this sequence has a limit $a$, i.e., $\lim _{n\to \infty }=a,$ if for any $\varepsilon >0$, there exists a number $N$, such that $|a_{n}-a|<\varepsilon$ for every $n>N$. Intuitively, this means that if you have an infinitely small ball centered at the limit point, then most of the points of the sequence are in that ball.

Limit of a Function

Let $f(x)$ be a real valued function in one variable. We say that

$\lim _{x\to p}f(x)=L$

if for any error $\varepsilon$, we can find a sufficiently small neighborhood of $p$ so that $f(x)$ is within $\varepsilon$ of the value $L$ for any $x\neq p$ in that neighborhood. One says that the limit of f(x) at p exists and is equal to L.

The standard, though more verbose, way of saying this is that for any $\varepsilon >0$ there exists a sufficiently small $\delta >0$ such that $|f(x)-L|<\varepsilon$ whenever $0<|x-p|<\delta$.

Intuitively, this means that as the variable $x$ approaches the value $p$, the function $f(x)$ tends to the value $L$.