Chapter One Section Three Evaluating limits Analytically

Chapter One Section Three So, what is a limit anyway? Below are four mathematical statements that are all correct. We will discuss, for each one, why it is correct and what it means for us.

Chapter One Section Three How do we even read such a statement? The first one can be read as  The limit of x x + 3 is 18 as x approaches 3  We could, alternatively, say that as x approaches 3, then x x + 3 approaches 18. What we are concerned with here is to discuss tendencies of the function (its y - values) in the target neighborhood identified for its x-values. We look first at what x is doing, then decide what the function is doing.

Chapter One Section Three Think about those other statements for a little while to convince yourself they are true. Maybe look at graphs, maybe try to simplify the expressions algebraically, maybe you can look at tables of values to convince yourself. Try not to move on in this slide show until you can convince yourself of at least one of the others.

Chapter One Section Three Your text provides some nice discussion as well, but it also provides some formal discussion that is beyond what we will do in this course. The discussion on pages 52 – 54 do not concern you this year.

Chapter One Section Three Let’s look at the algebra involved in the most difficult of the four examples. When presented with the expression:  First, we should examine this with simple arithmetic. Can we evaluate what we have been asked to evaluate?

Chapter One Section Three No, we cannot. We end up with the fraction of 0 divided by 0. Here, our first guess might be to say that this expression cannot be evaluated. A simple graph and table helps here.

Chapter One Section Three The evidence from the previous slide suggests that there is some predictable behavior for this function in the neighborhood as x approaches 0. A closer table look backs up this assumption. Do you know how to generate the following table?

Chapter One Section Three Unfortunately, this does not look like the answer we were given back on slide #2 We need to remember some algebra here and we need to apply it in an unusual way. If the original expression had been presented with the radicals in the denominator, would you have had a reaction telling you how to rewrite the fraction?

Chapter One Section Three That reaction will work here. Try to remember how to rationalize, and we will work on this together in class. Be prepared to share what progress you made on this and the other expressions presented on slide #2.

Chapter One Section Three What we are trying to develop here is a comfort level with algebraically manipulating these expressions. Remember, we are not concerned with what happens at the instant that x arrives at its target value. We want to discuss the tendency of the y -values as x approaches its target limit value.