Limits at Infinity - Practice Problems Solutions

Limits at Infinity - Practice Problems Solutions - Calculus...

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Calculus I © 2007 Paul Dawkins 1 http://tutorial.math.lamar.edu/terms.aspx Preface Here are the solutions to the practice problems for my Calculus I notes. Some solutions will have more or less detail than other solutions. The level of detail in each solution will depend up on several issues. If the section is a review section, this mostly applies to problems in the first chapter, there will probably not be as much detail to the solutions given that the problems really should be review. As the difficulty level of the problems increases less detail will go into the basics of the solution under the assumption that if you’ve reached the level of working the harder problems then you will probably already understand the basics fairly well and won’t need all the explanation. This document was written with presentation on the web in mind. On the web most solutions are broken down into steps and many of the steps have hints. Each hint on the web is given as a popup however in this document they are listed prior to each step. Also, on the web each step can be viewed individually by clicking on links while in this document they are all showing. Also, there are liable to be some formatting parts in this document intended for help in generating the web pages that haven’t been removed here. These issues may make the solutions a little difficult to follow at times, but they should still be readable. Limits At Infinity, Part I 1. For ( ) 73 4 1 89 f xxx =-+ evaluate each of the following limits. (a) ( ) lim x fx fi-¥ (b) ( ) lim x fi¥ (a) ( ) lim x To do this all we need to do is factor out the largest power of x from the whole polynomial and then use basic limit properties along with Fact 1 from this section to evaluate the limit. () ( ) 7 37 47 7 1 lim 4 1 8 9 li m4 1 li m li m 44 xx x x fi- ¥ ¥ Øø - + = -+ ²³ Œœ Ll ºû = - + = - ¥ = -¥ (b) ( ) lim x
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Calculus I © 2007 Paul Dawkins 2 http://tutorial.math.lamar.edu/terms.aspx For this part all of the mathematical manipulations we did in the first part did not depend upon the limit itself and so don’t need to be redone here. We can pick up the problem right before we actually took the limits and then proceed. () ( ) ( ) 7 37 47 1 89 lim 4 1 li m lim 44 x xx x ¥ ¥ fi¥ Øø - + = - + = ¥ ²³ Œœ Ll ºû 2. For ( ) 2 3 1 22 h t t tt =+- evaluate each of the following limits. (a) ( ) lim t ht fi-¥ (b) ( ) lim t (a) ( ) lim t To do this all we need to do is factor out the largest power of x from the whole polynomial and then use basic limit properties along with Fact 1 from this section to evaluate the limit. Note as well that we’ll convert the root over to a fractional exponent in order to allow it to be easier to deal with. Also note that this limit is a perfectly acceptable limit because the root is a cube root and we can take cube roots of negative numbers! We would only have run into problems had the index on the root been an even number.
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This note was uploaded on 02/05/2011 for the course CALCULUS/C 202 taught by Professor Tadius during the Spring '11 term at Benedictine KS.

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Limits at Infinity - Practice Problems Solutions - Calculus...

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