Techniques for Evaluating Limits. Techniques for Evaluating Limits. Techniques for Evaluating Limits

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1 PreCalc CH 12.2.notebook In the beginning of this chapter we evaluated limits of rational functions and saw that the indeterminate form gave us problems. There is a technique that can overcome this difficulty and the theorem on the next slide allows use to simplify rational functions so that we can eliminate the indeterminate form when evaluating limits. Functions That Agree At All But One Point Let c be a real number and let f(x) = g(x) for all x c in an open interval containing c. If the limit of g(x) as x approaches c exists, then the limit of f(x) also exists and: If we can factor a rational function, and then cancel terms, we have two functions that agree at all but one point. Brief review of factoring. Factor: We can make a table and find the two values that multiply to 15 and add up to 8. add to 8 mult. to = = 8 1, 15 3, 5 Using direct substitution on the limit below, we get the indeterminate form. Now we will factor the numerator. Then cancel terms and then use direct substitution. 1

2 PreCalc CH 12.2.notebook More Examples of Functions That Agree At All But One Point. More Examples of Functions That Agree At All But One Point. In the last three examples, we have seen a term in the denominator that seemed as though it would have created an asymptote, but it did not. Sometimes after factoring, variables remain in the denominator, so asymptotes will exist, however they only become important if we are looking for the limit of the function at the value that makes the denominator equal to zero. 2

3 PreCalc CH 12.2.notebook In this case, we can still find the limit, but what would happen if we wanted to find the limit as x 2? The graph on the next slide will show you why the limit exists at x= 2 and the limit does not exist at x=2. Strategies for Finding Limits 1. Recognize which limits can be evaluated using direct substitution. 2. If the limit of a function can not be evaluated as x approaches c, factor the function, cancel terms and then evaluate using direct substitution. 3. Know when the limit does not exist. 4. Use a table or graph to confirm your conclusion about the limit of a function. Find the limit. This example is exactly the same as the previous one, except x is now approaching 4. 3

4 PreCalc CH 12.2.notebook Rationalizing Technique When you have square roots, you are going to need to multiply by their conjugates. 4

5 PreCalc CH 12.2.notebook One Sided Limits We saw one sided limits in the beginning of this chapter, but did not call them one sided limit. In the first example of when limits do not exist, we looked at the behavior of the limit below. x f(x) ? The behavior differs from the left as x approaches zero as compared to x approaching zero from the right. This is how one sided limits are written for this example. The limit as x approaches from the left. The limit as x approaches from the right. 5

6 PreCalc CH 12.2.notebook Existence of a Limit If is a function and are real numbers, then if and only if both the left and right limits exist and are equal to. Find the limit Find the limit 6

7 PreCalc CH 12.2.notebook Find the limit Find the limit Find the limits 7

8 PreCalc CH 12.2.notebook Limits of a Difference Quotient Lets review difference quotients prior to fining their limits. Evaluate the following function. Now evaluate the difference quotient. Now find the limit of the difference quotient. Using the same function as the previous example, find the limit of the difference quotient. 8

9 PreCalc CH 12.2.notebook First we will evaluate: 9