Notes, Lesson 2.3
Calculating Limits Using the Limit Laws

Limit Laws | Check Concepts | Limit Techniques | Example Problems

We have seen that the calculation of limits is not reliable. We need some algebraic techniques for finding limits. We first must learn the laws of limits: There is good news. As you look at each of these laws, I believe that you will agree with me that these laws are intuitive. They behave in a way that we would all want them to behave.

Limit Laws
 Limit Law Verbal Description of Law When x is approaching the value of c, there is no effect whatsoever on b. This limit is evaluated by using the definition of the  notation . If an exponential expression has its base approaching a limit, the exponential expression will approach the limit to the same power. The root of a number which is approaching a limit is the root of the limit. The limit of a constant times a function is the constant times the limit of the function. The limit of a sum is the sum of the limits, and  The limit of a difference is the difference of the limits. The limit of a product is the product of the limits. The limit of a quotient is the quotient of the limits, if the limit in the denominator is not equal to zero. The limit of a function to a power can be calculated by taking the power of the function and then taking the limit, or by taking the limit and then raising the limit to the same power. This is really identical to the property immediately above, with the power being a fraction.

Limit Techniques
 # Technique for calculating: 1 Try substituting a into the limit expression. If you can solve this expression, you're done. This technique only works when the function is continuous at x = a. 2 If after you substitute, you can't simplify, try simplifying algebraically first, then substitute. 3 If you get a number for the numerator and a zero in the denominator, then there is no limit. 4 If you get zero in the numerator and the denominator, keep working, there is a limit. 5 To evaluate the limit of a rational function at infinity, divide numerator and denominator by the highest power of x that shows up in the denominator. Then subsitute to find the limit.

Example Problem:

 Given example problem. Direct substitution of 1 for x fails. When substituting, we get 0/0 which is of course impossible, and yields no information about the limit we are trying to calculate. The 0/0 result from substituting tells us that there must be a limit, therefore we.... We try technique #2, and we first attempt to algebraically simplify. Here we factor the numerator and cancel. Now that we've simplified, we try substitution again. This works, and we are done.

Example Problem:

 Given example problem. Direct substitution of 0 for t fails. When substituting, we get 0/0 which is of course impossible, and yields no information about the limit we are trying to calculate. The 0/0 result from substituting tells us that there must be a limit, therefore we.... We then work on simplifying algebraically. Here multiplying the numerator and denominator by the conjugate of the numerator works well. Simplifying after this multiplication gets us to this point. And now substitution works just fine.

Limits Teaching Video
 If you have the Journey Through Calculus CD, load and run MResources/Module 2/The Essential Examples/Examples D and E. This module will allow you to explore limits interactively.

 If you have the Journey Through Calculus CD, load and run MResources/Module 2/The Essential Examples/Example C . This module will allow you to explore limits interactively.

 If you have the Journey Through Calculus CD, load and run MResources/Module 2/Basics of Limits/Sound of a Limit that Exists. This module will allow you to watch an animation of a limit.

Calculating Limits @ Calculus-Help.com
 Check Concepts

 #1: True or False. The limit of a function as it approaches 1 has nothing to do with the value of the function at 1. Choose One True False #2: True or False. A left-hand limit is the limit of the function as it heads toward the left. Choose One True False #3: True or False. The limit laws are intuitive. Choose One True False #4: True or False. To find a limit, both the left and right hand limits must agree. Choose One True False #5: True or False. The limit of a discontinuous function does not exist. Choose One True False