This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Limits At Infinity, Part II In the previous sec tion we looked at limits at infinity of polynomials and/or rational expression involving polynomials. In this section we want to take a look at some other types of functions that often show up in limits at infinity. The functions well be looking at here are exponentials, natural logarithms and inverse tangents. Lets start by taking a look at a some of very basic examples involving exponential functions. Example 1 Evaluate each of the following limits. Solution There are really just restatements of facts given in the basic exponential section of the review so well leave it to you to go back and verify these. The main point of this example was to point out that if the exponent of an exponential goes to infinity in the limit then the exponential function will also go to infinity in the limit. Likewise, if the exponent goes to minus infinity in the limit then the exponential will go to zero in the limit. Heres a quick set of examples to illustrate these ideas. Example 2 Evaluate each of the following limits. (a) [ Solution ] (b) [ Solution ] (c) [ Solution ] Solution (a) In this part what we need to note (using Fact 2 above) is that in the limit the exponent of the exponential does the following, So, the exponent goes to minus infinity in the limit and so the exponential must go to zero in the limit using the ideas from the previous set of examples. So, the answer here is, [ Return to Problems ] (b) Here lets first note that, The exponent goes to infinity in the limit and so the exponential will also need to go to infinity in the limit. Or, [ Return to Problems ] (c) On the surface this part doesnt appear to belong in this section since it isnt a limit at infinity. However, it does fit into the ideas were examining in this set of examples. So, lets first note that using the idea from the previous section we have, Remember that in order to do this limit here we do need to do a right hand limit. So, the exponent goes to infinity in the limit and so the exponential must also go to infinity. Heres the answer to this part. [ Return to Problems ] Lets work some more complicated examples involving exponentials. In the following set of examples it wont be that the exponents are more complicated, but instead that there will be more than one exponential function to deal with. Example 3 Evaluate each of the following limits. (a) [ Solution ] (b) [ Solution ] Solution So, the only difference between these two limits is the fact that in the first were taking the limit as we go to plus infinity and in the second were going to minus infinity. To this point weve been able to reuse work from the first limit in the at least a portion of the second limit. With exponentials that will often not be the case, were going to treat each of these as separate problems....
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
 sc
 Polynomials, Limits

Click to edit the document details