The following problems require the use of the precise 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 definition of the limit of a function as x approaches a constant.

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

if , then .

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

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

We first pick an band around the number *L* on the *y*-*axis* . We then determine a band around the number *a* on the *x*-*axis* so that for all *x*-*values* (excluding *x*=*a* ) inside the band, the corresponding *y*-*values* lie inside the band. In other words, we first pick a prescribed closeness () to *L* . Then we get close enough () to *a* so that all the corresponding *y*-*values* fall inside the band.
If a can be found for each value of , then we have proven that
*L* is the correct limit. If there is a single 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 be given. Then take the expression and, from this, attempt to algebraically ``solve for" | *x* - *a* | . At that point, an appropriate value for 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.

*PROBLEM 1 :*Prove that .Click HERE to see a detailed solution to problem 1.

*PROBLEM 2 :*Prove that .Click HERE to see a detailed solution to problem 2.

*PROBLEM 3 :*Prove that .Click HERE to see a detailed solution to problem 3.

*PROBLEM 4 :*Prove that .Click HERE to see a detailed solution to problem 4.

*PROBLEM 5 :*Prove that .Click HERE to see a detailed solution to problem 5.

*PROBLEM 6 :*Prove that .Click HERE to see a detailed solution to problem 6.

*PROBLEM 7 :*Prove that .Click HERE to see a detailed solution to problem 7.

*PROBLEM 8 :*Prove that .Click HERE to see a detailed solution to problem 8.

*PROBLEM 9 :*Prove that .Click HERE to see a detailed solution to problem 9.

*PROBLEM 10 :*Prove that .Click HERE to see a detailed solution to problem 10.

*PROBLEM 11 :*Prove that .Click HERE to see a detailed solution to problem 11.

*PROBLEM 12 :*Prove that .Click HERE to see a detailed solution to problem 12.

*PROBLEM 13 :*Prove that , where*a*is any real number .Click HERE to see a detailed solution to problem 13.

*PROBLEM 14 :*Prove that , where*a*is any positive real number .Click HERE to see a detailed solution to problem 14.

for any two real numbers *A* and *B* .

*PROBLEM 15 :*Let . Prove that does not exist .Click HERE to see a detailed solution to problem 15.

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

Wed Apr 30 16:21:53 PDT 1997