What do we mean by ``approaching''?

Consider the following graph for $f:[-3,3]\rightarrow(-\infty,\infty)$:

We would like to discuss what we mean by the behaviour of f(x) as x approaches various values of $a\in[-3,3]$:

x approaches -3:

As x approaches -3 it seems natural to say that f(x) approaches -2. From this we infer that by the phrase ``x approaches -3 we mean that x gets as close to -3 as we like. Else, we could say that if x approaches -5/2 through values larger than -5/2 then x approaches -3. This is not at all what we have in mind.

x approaches -1:

Here we seem to have a dilemna: If x approaches -1 through values less than -1 the value of f(x) approaches -1, and if it approaches through values of x which are larger than -1 then f(x) approaches 1. What is worse, f(-1) = 0. In this case we would be forced to conclude that as x approaches -1 there is no single value that f(x) approaches. From this we conclude that we must mean by f(x) approaches b'' that b is the only number that f(x) approaches.

x approaches 1:

Here we have removed one of the dilemmas of the previous case. Now the only problem is that f(1) is not equal to , the value that f(x) approaches. This is all right, insofar as we are not interested in what the value of f is at 1, since in many cases we have seen, the function isn't even defined at the value that its variable approaches.

So what is the big deal? Well, from experience we would bet that the rule for the function f above is not a polynomial. What is more, while -1 is a zero for f and 1 is not a zero for f, the bisection method for estimating zeroes would fail horribly for this f. One of the things we will have to distinguish is when will a function not have this strange behaviour, and instead have the behavious that as x approaches a the values f(x) approach f(a). Such functions will be called continuous.

In order to resolve this, we will need to find a proper definition for this ``approaching'' business. The formal study of this is the study of limits, which we shall take up in the next lecture.