PRECISE LIMITS OF FUNCTIONS AS X APPROACHES A CONSTANT


The following problems require the use of the precise tex2html_wrap_inline79 definition of limits of functions as x approaches a constant. Most problems are average. A few are somewhat challenging. We will begin with the precise tex2html_wrap_inline79 definition of the limit of a function as x approaches a constant.


DEFINITION: The statement tex2html_wrap_inline83 has the following precise definition. Given any real number tex2html_wrap_inline85 , there exists another real number tex2html_wrap_inline87 so that

if tex2html_wrap_inline89 , then tex2html_wrap_inline91 .


In general, the value of tex2html_wrap_inline93 will depend on the value of tex2html_wrap_inline95 . That is, we will always begin with tex2html_wrap_inline85 and then determine an appropriate corresponding value for tex2html_wrap_inline87. There are many values of tex2html_wrap_inline93 which work. Once you find a value that works, all smaller values of tex2html_wrap_inline93 also work.


To try and understand the meaning behind this abstract definition, see the given diagram below.






We first pick an tex2html_wrap_inline95 band around the number L on the y-axis . We then determine a tex2html_wrap_inline93 band around the number a on the x-axis so that for all x-values (excluding x=a ) inside the tex2html_wrap_inline93 band, the corresponding y-values lie inside the tex2html_wrap_inline95 band. In other words, we first pick a prescribed closeness (tex2html_wrap_inline95) to L . Then we get close enough (tex2html_wrap_inline93) to a so that all the corresponding y-values fall inside the tex2html_wrap_inline95 band. If a tex2html_wrap_inline87 can be found for each value of tex2html_wrap_inline85, then we have proven that L is the correct limit. If there is a single tex2html_wrap_inline85 for which this process fails, then the limit L has been incorrectly computed, or the limit does not exist.


In the problems that follow, we will use this precise definition to mathematically PROVE that the limits we compute algebraically are correct. When using this definition, begin each proof by letting tex2html_wrap_inline85 be given. Then take the expression tex2html_wrap_inline91 and, from this, attempt to algebraically ``solve for" | x - a | . At that point, an appropriate value for tex2html_wrap_inline87 can easily be determined.

The expression `` iff " will be used often in the solutions to the following problems. It means `` if and only if " or `` is equivalent to ''. The expression `` min{A, B }" will also be used in many of the solutions. It means `` the minimum value of A and B." For example, min{ 3, 7 } = 3.


The following two problems require some knowledge and understanding of the Mean Value Theorem.


The following problem uses the triangle inequality. The triangle inequality states that

tex2html_wrap_inline189

for any two real numbers A and B .

Click HERE to return to the original list of various types of calculus problems.

Your comments and suggestions are welcome. Please e-mail any correspondence to Duane Kouba by clicking on the following address :

kouba@math.ucdavis.edu

Duane Kouba
Wed Apr 30 16:21:53 PDT 1997