On-line Math 21

1.3.1  Examples of computing limits using rules and theorems.

Problems with hints

Problem 1 Compute

lim x® 3  x2-9 x2-2x-3 .

How do I get started?

The problem with this limit is that, as x approaches 3 , both the numerator and the denominator approach 0. So, the fraction becomes

0 0

in the limit. This is what is called later on an indeterminate form. For now, though, we need to make sense of this particular 0/0 - not the general idea of what 0/0 ``should'' be, but what we mean by it in this case. Here, we have a limit, so this 0/0 ought to be the limit of the values of the expression as x approaches 3 .

We want to use the rather obvious property that if two function are the same, except at 3 (say), then their limits at 3 will also be the same. In this case, that means we have to factor the numerator and the denominator, and see if that will let us make sense of the fraction for numbers near (maybe even at) x = 3 . You should be able to see why the 0/0 happens once you factor the expression.

Did I factor it correctly?

What's the answer?

By David L. Johnson, last modified 3/10/00.