Clearest definition of a limit

Question

What is the most elegant and clearest definition of limit that you know?

For me is this:

Let $f(x)$ be a function defined on an interval that contains $a$, except possibly at $a$. Then we say that:

$\lim_{x\to a}f(x)=l$

if for every number $\varepsilon>0$ there is some number $\delta>0$ such that:

$\lvert f(x)-l\rvert<\varepsilon $

whenever:

$0<\lvert x-a\rvert<\delta $

Answer 1

It is just the usual definition for finite limit $l$ with x which tends to a finite cluster point $a$.

Note that as an alternative someone set that $x\neq a$, in this case it suffices that $\lvert x-a\rvert<\delta$.

Answer 2

I am a bit biased, but my favourite definition of the limit is from nonstandard analysis.

Let $f(x)$ be a function on a punctured interval $(r, s) \setminus \{a\}$. Let $^*f$ be the transfer of this function to the hyperreals. We say $\lim_{x \to a} f(x) = l$ if, whenever $x$ is infinitesimally close to $a$, $^*f(x)$ is infinitesimally close to $l$.

Answer 3

$\lim_{x\to a}f(x)=l$ if by restricting $f$ to an appropriate punctured neighbourhood of $a$ you can fit its image into any neighbourhood of $l$.

• This works for every topological space, not just for real numbers.

• It uses terms whose meaning for real numbers (and some more general spaces) is intuitively easy to grasp, and therefore supports both an intuitive and a rigorous level.