This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: Infinite Limits In this section we will take a look at limits whose value is infinity or minus infinity. These kinds of limit will show up fairly regularly in later sections and in other courses and so you’ll need to be able to deal with them when you run across them. The first thing we should probably do here is to define just what we mean when we sat that a limit has a value of infinity or minus infinity. Definition We say if we can make f(x) arbitrarily large for all x sufficiently close to x=a , from both sides, without actually letting . We say if we can make f(x) arbitrarily large and negative for all x sufficiently close to x=a , from both sides, without actually letting . These definitions can be appropriately modified for the one-sided limits as well. To see a more precise and mathematical definition of this kind of limit see the The Definition of the Limit section at the end of this chapter. Let’s start off with a fairly typical example illustrating infinite limits. Example 1 Evaluate each of the following limits. Solution So we’re going to be taking a look at a couple of one-sided limits as well as the normal limit here. In all three cases notice that we can’t just plug in . If we did we would get division by zero. Also recall that the definitions above can be easily modified to give similar definitions for the two one-sided limits which we’ll be needing here. Now, there are several ways we could proceed here to get values for these limits. One way is to plug in some points and see what value the function is approaching. In the proceeding section we said that we were no longer going to do this, but in this case it is a good way to illustrate just what’s going on with this function. So, here is a table of values of x ’s from both the left and the right. Using these values we’ll be able to estimate the value of the two one-sided limits and once we have that done we can use the fact that the normal limit will exist only if the two one-sided limits exist and have the same value. x x-0.1-10 0.1 10-0.01-100 0.01 100-0.001-1000 0.001 1000-0.0001-10000 0.0001 10000 From this table we can see that as we make x smaller and smaller the function gets larger and larger and will retain the same sign that x originally had. It should make sense that this trend will continue for any smaller value of x that we chose to use. The function is a constant (one in this case) divided by an increasingly small number. The resulting fraction should be an increasingly large number and as noted above the fraction will retain the same sign as x . We can make the function as large and positive as we want for all x ’s sufficiently close to zero while staying positive ( i.e. on the right). Likewise, we can make the function as large and negative as we want for all x ’s sufficiently close to zero while staying negative ( i.e. on the left)....
View Full Document
This note was uploaded on 11/06/2011 for the course MATH 151 taught by Professor Sc during the Fall '08 term at Rutgers.
- Fall '08