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EVALUATING LIMITS ANALYTICALLY Section 1.3. When you are done with your homework, you should be able to… Evaluate a Limit Using Properties of Limits Develop.

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Presentation on theme: "EVALUATING LIMITS ANALYTICALLY Section 1.3. When you are done with your homework, you should be able to… Evaluate a Limit Using Properties of Limits Develop."— Presentation transcript:

1 EVALUATING LIMITS ANALYTICALLY Section 1.3

2 When you are done with your homework, you should be able to… Evaluate a Limit Using Properties of Limits Develop and Use a Strategy for Finding Limits Evaluate a Limit Using Dividing Out and Rationalizing Techniques Evaluate a Limit Using the Squeeze Theorem

3 SOME BASIC LIMITS Let and b and c be real numbers and let n be a positive integer

4 Evaluate A.5 B.-3 C.Does not exist

5 Evaluate A.1 B.-1 C.5 D.Does not exist

6 Properties of Limits Let and b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits: 1.Scalar multiple 2.Sum or difference 3.Product 4.Quotient 5.Power

7 Evaluate A.-4 B.-2 C.1 D.Does not exist

8 Limits of Polynomial and Rational Functions If p is a polynomial function and c is a real number, then. If r is a rational function given by and c is a real number such that then

9 Evaluate A.5 B.0 C.-5 D.Does not exist

10 Evaluate the function at x = 2 A.1 B.DNE

11 The Limit of a Function Involving a Radical  Let n be a positive integer. The following limit is valid for all c if is n odd, and is valid for if n is even.

12 The Limit of a Composite Function Let f and g be functions with the following limits: Then

13 Limits of Trigonometric Functions Let c be a real number in the domain of the given trigonometric function.

14 STRATEGIES FOR FINDING LIMITS Functions That Agree at All But One Point –Let c be a real number and let for all x in an open interval containing c. If the limit of g as x approaches c exists, then the limit of f also exists and.

15 A Strategy for Finding Limits Learn to recognize which limits can be evaluated by direct substitution. If the limit of f(x) as x approaches c cannot be evaluated by direct substitution, try to find a function g that agrees with f for all x other than x = c. Apply. Use a graph or table to reinforce your conclusion.

16 Dividing Out Techniques Example:

17 Evaluate. A.0 B.Does not exist C.1/16

18 Rationalizing Techniques Example:

19 Evaluate the exact limit.

20 TWO SPECIAL TRIGONOMETRIC LIMITS

21 Evaluate the exact answer.


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