Limits and Curve Sketching

By definition, lim_x→c f(x) = L means that for every hyperreal number x which is infinitely close but not equal to c, f(x) is infinitely close to L. What does lim_x→c f(x) = L tell us about f(x) for real numbers x? It turns out that if lim_x→c f(x) = L, then for every real number x which is close to but not equal to c, f(x) is close to L.

In the next section we shall justify the above intuitive statement by a mathematical theorem. The main difficulty is to make the word "close" precise. For the time being we shall simply illustrate the idea with some examples.

Example 1

Other types of limits also give information which is useful in drawing graphs. For instance, if lim_x→c f(x) = ∞, then for every number x which is close to but not equal to c, the value of f(x) is large. And if lim_x→∞ f(x) = L, then for every large real number x, f(x) is close to L.

In both the above statements, if we replace "close" by "infinitely close" and "large" by "infinitely large" we get our official definition of a limit. We give two more examples.

Example 2

Example 3

In Chapter 3 we showed how to use the first and second derivatives to sketch the graph of a function which is continuous on a closed interval. In the next example we shall sketch the graph of the function f(x) = 1 + 1/(x - 2)². But this time the function is discontinuous at x = 2, and the domain is the whole real line except for the point x = 2. Our method uses not only the values but also the limits of the function and its first derivative.

Example 4

Suppose the function f and its derivative f' exist and are continuous at all but a finite number of points of an interval I. The following procedure can be used in sketching the curve y = f(x).

We shall now work several more examples; the steps in computing the limits are left to the student.

Example 5

Example 6

Example 7