# Functions, Limits, and Continuity

### Contents

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#### Limits and Continuity

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 x0 be a real number. We say that f (x) has a limit at x = x0 if as x approaches x0, f (x) approaches some number L. We call the number L the limit of the function f (x) at x0, 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 = x0 only if x approaches x0 from one side or the other. In fact, the function may have two different limits at x0, depending on the side from which x approaches x0. Such limits are called one-sided limits. If f has the limit L when x approaches x0 from the left, we say f has left-hand limit L at x0 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 x0, and L = f (x0), then f is said to be continuous at x0. Note that this presupposes that x0 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.