# Limit of a function

Hi
What is the definition of limit of a function

There are a couple of ways for defining the limit of a function f(x) say as x-->a, where a could be a real nr. or infinity.
One of these ways is using $$\epsilon,\delta$$

Def.Let f(x) be a function defined in an open interval containing a. Then A is said to be the limit of the function f(x) as x goes to a, if for every $$\epsilon>0,\exists \delta(\epsilon)>0$$ such that

$$|f(x)-A|<\epsilon, / / / / whenever / / / / 0<|x-a|<\delta$$

and we write it: $$\lim_{x\rightarrow a}f(x)=A$$

Another way to define it, is using heine's definition using sequences, then another way is in terms of infinitesimals.

Note: the reason that it is required that $$0<|x-a|$$ is that it is not necessary for the function f to be defined at x=a, since when working with the limit as x-->a we are not really interested what happens exactly at x=a, but rather how the function behaves in a vicinity of a.

Hi
What is the definition of limit of a function

If $$f$$ is a function and $$\epsilon$$ is an infinitesimal, the real part of $$f(x+\epsilon)$$ is the limit as f approaches x.

It is useful for when a function with "holes" in them, as well as functions which jump up infinitely high when they are evaluated close to a point. For example,

$$f(x) = \frac{x^3}{x}$$

is a function which is no defined at x=0. If you graph the function, it looks *exactly* the same as x^2, except that there is a "hole" at the origin. Since f(0) = 0^3 / 0 = 0/0, it is undefined.

Taking the limit:

$$\lim_{x->0} f(x)$$

allows us to ignore this illegal move, and give us a well-defined answer that is "for all practical purposes" equivalent.

Limits crop up everywhere in calculus. The definition of a derivative, for example is:

$$f'(x) = \lim_{h-> 0} \frac{f(x+h) - f(x)}{h}$$

If you were to try an evaluate $$\frac{f(x+h) - f(x)}{h}$$ with h = 0, you'd get an undefined answer. Taking the limit instead allows us to get a useful answer.