The *limit* of a function is that value which the function approaches as x approaches the given value of a.

**Definition:** A function f(x) is *continuous* at x = a if all three of the following conditions are satisfied:

- f(a) exists
- exists
- f(a) =

Given the following function f(x):

- f(-3) exists and equals the y value which is -2
- exists and equals the y value which is -2 because the function approaches -2 as you approach x = -3 from the left and right
- because -2 = -2

So f(x) is continuous at x = -3

**At the point x = -1:**

1. f(-1) does not exist

So f(x) is **not** continuous at x = -1

**At the point x = 1:**

- f(1) exists and equals the y value which is 0
- does not exist because the function approaches 2 from the right and 0 from the left

In order for to exist you must approach the same number
from the right and left. Since you get different values the does not exist.

So f(x) is not continuous at x = 1.

Other Examples:

Evaluate each of the following limits

Problem 1

Problem 2

does not exist

Problem 3

Problem 4

Problem 5

*Copyright © 1996 by B. Chambers and P. Lowry. All Rights Reserved.*