lim x → 2 g x , lim x → 2 f x , lim x → c 1 f x lim x → c 5 g x 4 f x lim x → c 6 f x g x lim x → c f x g x 2 lim x → c g x 2 lim x → c f x 5, lim x → c f x lim x → c f x g x lim x → c f x g x lim x → c 2 g x lim x → c g x 6 lim x → c f x 3, lim x → 1 f x f x ln 7 x , lim x → 4 f x f x ln x 3 , lim x → 3 f x f x 7 x 3 , lim x → 1 f x f x x 1 x 2 4 x 3 , lim x → 2 f x f x x 5 4 x 2 , lim x → 4 f x f x x 3 1 x 4 , lim x → 1 f x f x sin x , lim x → 0 f x f x cos 1 x , lim x → 0 f x f x e x 1 x , lim x → 0 f x f x 5 2 e 1 x ,
11.2 Techniques for Evaluating Limits What you should learn Use the dividing out technique to evaluate limits of functions. Use the rationalizing technique to evaluate limits of functions. Approximate limits of functions graphically and numerically. Evaluate one-sided limits of functions. Evaluate limits of difference quotients from calculus. Why you should learn it Many definitions in calculus involve the limit of a function. For instance, in Exercises 77 and 78 on page 799, the definition of the velocity of a free-falling object at any instant in time involves finding the limit of a position function. Section 11.2 Techniques for Evaluating Limits 791 Dividing Out Technique In Section 11.1, you studied several types of functions whose limits can be evaluated by direct substitution. In this section, you will study several techniques for evaluating limits of functions for which direct substitution fails. Suppose you were asked to find the following limit. Direct substitution fails because is a zero of the denominator. By using a table, however, it appears that the limit of the function as is Another way to find the limit of this function is shown in Example 1. 5. x → 3 3 lim x → 3 x 2 x 6 x 3 Example 1 Dividing Out Technique Find the limit: Solution Begin by factoring the numerator and dividing out any common factors. Factor numerator. Divide out common factor. Simplify. Direct substitution Simplify. Now try Exercise 7. 5 3 2 lim x → 3 x 2 lim x → 3 x 2 x 3 x 3 lim x → 3 x 2 x 6 x 3 lim x → 3 x 2 x 3 x 3 lim x → 3 x 2 x 6 x 3 . This procedure for evaluating a limit is called the dividing out technique. The validity of this technique stems from the fact that if two functions agree at all but a single number c , they must have identical limit behavior at In Example 1, the functions given by and agree at all values of other than So, you can use to find the limit of f x . g x x 3. x g x x 2 f x x 2 x 6 x 3 x c . Peticolas Megna/Fundamental Photographs x 3.01 3.001 3.0001 3 2.9999 2.999 2.99 x 2 x 6 x 3 5.01 5.001 5.0001 ? 4.9999 4.999 4.99 Prerequisite Skills To review factoring techniques, see Appendix G, Study Capsule 1.
The dividing out technique should be applied only when direct substitution produces 0 in both the numerator and the denominator. The resulting fraction, has no meaning as a real number. It is called an indeterminate form because you cannot, from the form alone, determine the limit. When you try to evaluate a limit of a rational function by direct substitution and encounter this form, you can conclude that the numerator and denominator must have a common factor. After
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