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Limits and Continuity

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In this section we introduce the concepts of limit and continuous function.

Intuitive Definitions

Let f (x) be a function from a subset of the real numbers to the real numbers, and let x 0 be a real number. We say that f (x) has a limit at x = x 0 if as x approaches x 0 , f (x) approaches some number L . We call the number L the limit of the function f (x) at x 0 , and we write

f (x) = L    

If f is the function whose graph is drawn below, then f (x) = 2 and f (x) = 5 . Note that a function need not be defined at a particular value of x (that is, x need not belong to the domain) in order for a limit to exist there.

Figure %: Limit Examples

It may be that a function has a limit at x = x 0 only if x approaches x 0 from one side or the other. In fact, the function may have two different limits at x 0 , depending on the side from which x approaches x 0 . Such limits are called one-sided limits. If f has the limit L when x approaches x 0 from the left, we say f has left-hand limit L at x 0 and write

f (x) = L    

We make the similar definition for the right-hand limit. An example of a function with one-sided limits is shown below. What is the left-hand limit of f at -2 ? The right-hand limit?

Figure %: An Example of a One-Sided Limit

If a function f has a limit L at x 0 , and L = f (x 0) , then f is said to be continuous at x 0 . Note that this presupposes that x 0 is in the domain of f , which is not necessary when discussing limits in general. A function that is continuous at every point in its domain is said to be a continuous function.