Consider the function . Let's study the behavior of this function as x approaches 0. By looking at the graph and table, you can see that as x approaches zero from either side, the values of f(x) increase without bound.

To indicate this behavior, we say that the limit as x approaches zero of f(x) is infinity.

However, infinity is not a number. So note that, although we say that the limit is infinity, actually a limit does not exist. Saying that the limit is infinity is expressing a special way in which the limit does not exist. That is, the value of f(x) can be made as large as we want by taking x-values close enough to zero.

Now, let's study the behavior of near x = 2. From the graph and the table, you can see that does not exist. As x approaches 2 from the right, f(x) increases without bound. So the limit as x approaches 2 from the right is infinity.

As x approaches 2 from the left, f(x) decreases without bound. So the limit as x approaches 2 from the left is negative infinity.

So, as x approaches 2, the limit of f(x) is neither infinity nor negative infinity.

Find the indicated limit. You can use your calculator to look at the graph or the table of values for this function. Click "Submit" when finished.

Find the indicated limit. You can use your calculator to look at the graph or the table of values for this function. You can also find this limit by studying the quotient for x values that are close to 2 but not equal to 2. Click "Submit" when finished.

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