Limit Properties - Practice Problems Solutions

# Limit Properties - Practice Problems Solutions - Preface...

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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. Limit Properties 1. Given ( ) 8 li m9 x fx =- , ( ) 8 m2 x gx = and ( ) 8 m4 x hx = use the limit properties given in this section to compute each of the following limits. If it is not possible to compute any of the limits clearly explain why not. (a) ( ) ( ) 8 lim 2 12 x f x - Øø ºû (b) ( ) 8 lim 36 x - (c) ( ) ( ) ( ) 8 lim x gxh x - (d) ( ) ( ) ( ) 8 lim x f x g x -+ Hint : For each of these all we need to do is use the limit properties on the limit until the given limits appear and we can then compute the value. (a) ( ) ( ) 8 lim 2 12 x f x - Here is the work for this limit. At each step the property (or properties) used are listed and note that in some cases the properties may have been used more than once in the indicated step.

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( ) ( ) ( ) ( ) ( ) () ( ) 8 88 lim 2 1 2 lim 2 lim 1 2 Property 2 2li m 12li m Property 1 29 1 2 4 Plug in values of limits 66 x xx f x h x f x hx f x ﬁﬁ - =- Ø øØ ø Øø º ûº û ºû = -- (b) ( ) 8 lim 36 x - Here is the work for this limit. At each step the property (or properties) used are listed and note that in some cases the properties may have been used more than once in the indicated step. ( ) ( ) 8 lim 3 6 lim 3 lim 6 Property 2 3li m lim 6 Property 1 3 4 6 Plug in value of limits & Property 7 6 x h x - Ø øØø º ûºû = (c) ( ) ( ) ( ) 8 lim x gxh x fx - Here is the work for this limit. At each step the property (or properties) used are listed and note that in some cases the properties may have been used more than once in the indicated step.
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Limit Properties - Practice Problems Solutions - Preface...

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