12.2 Techniques for Evaluating Limits

Techniques for Evaluating Limits

There are a few ways to evaluate limits. Based on the given function, you need to decide which technique to use. As we learned before, limits are very well behaved when it comes to manipulating the function. That means that you can multiply, divide, subtract, raise to a power, etc. without having to alter the function. This is one way that makes limits easier.

Limits are defined as:

Continuous functions are defined as:

THIS DEFINITION IS NOT IN THE BOOK!!

One way to evaluate limits is by direct substitution.

Direct Substitution

If you are given a function f(x) that is a simple polynomial then you can just plug c into it to find the limit f(c). This also works with rational functions when the denominator is not equal to 0. 

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Dividing Out Technique

NOTE: When using the dividing out technique, if you get 0 in the denominator, then the limit does not exist!

If we wanted to evaluate the limit above as x approaches -3, we would not be able to do that since the denominator would equal 0. However, if you graphed the function you would see that the limit is -5. This is when the dividing out technique can be used. 

(Divide out the x+3)

(Now you can use the direct substitution method)

The only graphical difference between  and  is the hole in the graph.

You should only use the dividing out technique when you get 0 on the top and on the bottom after substitution of c. Otherwise you should use a different technique. 

Rationalizing Technique

The rationalizing technique can be used in the same way as the dividing out technique. That is when you get 0 on the top and bottom. However, when you can't divide common factors, you use this over the divide out technique. 

When you plug in c for the initial limit you will see that you get the indeterminate form (0/0) but when you do the same for the limit that we rationalized you get 1/2. That is how the rationalization form is used. Remember when you multiply by the conjugate you get the difference of squares. {(a+b)(a-b) = (a² - b²)}