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