Use the dividing out technique to evaluate limits of functions



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Use the dividing out technique to evaluate limits of functions

  • Use the dividing out technique to evaluate limits of functions

  • Use the rationalizing technique to evaluate limits of functions

  • Use technology to approximate limits of functions graphically and numerically



Evaluate one-sided limits of functions





We have 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.

  • We have 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 –3 is a zero of the denominator. By using a table, however, it appears that the limit of the function as x approaches –3 is –5.

  • Direct substitution fails because –3 is a zero of the denominator. By using a table, however, it appears that the limit of the function as x approaches –3 is –5.



Find the limit.

  • Find the limit.

  • Solution:

  • Begin by factoring the numerator and dividing out any common factors.



  • (x – 2)

  • = –3 – 2

  • = –5



This procedure for evaluating a limit is called the dividing out technique.

  • This procedure for evaluating a limit is called the dividing out technique.

  • The validity of this technique stems from the fact that when two functions agree at all but a single number c, they must have identical limit behavior at x = c.

  • In Example 1, the functions given by

  • f (x) and g (x) = x – 2 agree at all values of x other than x = –3.

  • So, you can use g (x) to find the limit of f (x).



The dividing out technique should be applied only when direct substitution produces 0 in both the numerator and the denominator.

  • The dividing out technique should be applied only when direct substitution produces 0 in both the numerator and the denominator.

  • An expression such as 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.

  • 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 factoring and dividing out, you should try direct substitution again.





The limit of f (x) as x c does not exist when the function f (x) approaches a different number from the left side of c than it approaches from the right side of c.

  • The limit of f (x) as x c does not exist when the function f (x) approaches a different number from the left side of c than it approaches from the right side of c.

  • This type of behavior can be described more concisely with the concept of a one-sided limit.

  • f (x) = L1 or f (x)  L1 as xc

  • f (x) = L2 or f (x)  L2 as xc+



Find the limit as x  0 from the left and the limit as x  0 from the right for f (x) = .

  • Find the limit as x  0 from the left and the limit as x  0 from the right for f (x) = .

  • Solution:

  • From the graph of f, shown in Figure 11.15, you can see that f (x) = –2 for all x < 0.



So, the limit from the left is

  • So, the limit from the left is

  • = –2.

  • Because f (x) = 2 for all x > 0, the limit from the right is

  • = 2.







A Limit from Calculus In the next section, you will study an important type of limit from calculus—the limit of a difference quotient.

  • A Limit from Calculus In the next section, you will study an important type of limit from calculus—the limit of a difference quotient.



For the function given by f (x) = x2 – 1, find

  • For the function given by f (x) = x2 – 1, find

  • Solution:

  • Direct substitution produces an indeterminate form.



  • By factoring and dividing out, you obtain the following.



= (6 + h)

  • = (6 + h)

  • = 6 + 0

  • = 6

  • So, the limit is 6.




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