Limit of A Function - Motivation

Motivation

Imagine a person walking over a landscape represented by the graph of y = f(x). His horizontal position is measured by the value of x, much like the position given by a map of the land or by a global positioning system. His altitude is given by the coordinate y. He is walking towards the horizontal position given by x = p. As he gets closer and closer to it, he notices that his altitude approaches L. Say there's a wall there so he can't stand on that point exactly, but can still get arbitrarily close to it. If asked about the altitude of x = p, he would then answer L.

What, then, does it mean to say that his altitude approaches L? It means that his altitude gets nearer and nearer to L except for a possible small error in accuracy. For example, suppose a particular accuracy goal is set for our traveler: he must get within ten meters of L in altitude. He reports back that indeed he can get within ten meters of L, since he notes that when he is anywhere within fifty horizontal meters of p, his altitude is always ten meters or less from L.

The accuracy goal is then changed: can he get within one vertical meter? Yes. If he is anywhere within seven horizontal meters of p, then his altitude always remains within one meter from the target L. In summary, to say that the traveler's altitude approaches L as his horizontal position approaches p means that for every target accuracy goal, however small it may be, there is some neighborhood of p whose altitude fulfills that accuracy goal.

The initial informal statement can now be explicated:

The limit of a function f(x) as x approaches p is a number L with the following property: given any target distance from L, there is a distance from p within which the values of f(x) remain within the target distance.

This explicit statement is quite close to the formal definition of the limit of a function with values in a topological space.