You have seen how limits arise when you want to find the area of a circle by calculating the areas of inscribed and circumscribed polygons. Now, let's study limits in general and find ways to compute them.

Consider the function . To find the limit of the function as x approaches 3, we'll investigate the behavior of the function for values of x close to 3. When studying limits, the behavior of the function at x = 3 is not relevant.

Look at the graph. As x approaches 3 from the right, f(x) approaches -3.

Similarly, as x approaches 3 from the left, f(x) approaches -3.

In fact, the value of f(x) can be made as close to -3 as we want by making x sufficiently close to 3. For example, if x is within 0.01 unit of 3, the value of f(x) is within 0.04 unit of -3. We can make f(x) even closer to -3 by making x closer to 3.

We say that the limit of x 2 - 3x - 3 as x approaches 3 is -3.

Now consider the graph of g(x). We will explore the limit of this function at different x-values.

First, let's find the limit of g(x) as x approaches 1. If x is close to 1, but not equal to 1, g(x) is close to 4. So the limit of g(x) as x approaches 1 is 4. Notice that the fact that x cannot equal 1 does not affect the value of the limit.

Now let's find the limit of g(x) as x approaches 2.5. If x is close to 2.5, but not equal to 2.5, g(x) is close to 0.5. So the limit of g(x) as x approaches 2.5 is 0.5. Notice that the fact that x can equal 2.5 does not affect the value of the limit.

What is the limit of g(x) as x approaches 2? Notice that if x is close to 2 on the left, g(x) is close to 3. However, if x is close to 2 on the right, g(x) is close to 2.5. There is no unique number that g(x) can be kept close to just by keeping x close to 2. So the limit of g(x) as x approaches 2 does not exist. We write DNE to abbreviate "does not exist."

Finally, let's look at the limit of g(x) as x approaches 3. The values of g(x) can be made arbitrarily large by making x close enough to 3. So the values of g(x) do not approach a unique number. Therefore, the limit of g(x) as x approaches 3 does not exist.

Decide whether the function shown has a limit at each of the given x-values. If the limit exists, find what it equals. If it doesn't, write DNE for does not exist. Click "Submit" when finished.

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education