Section 12.2

1. Section 12.2 Techniques for Evaluating Limits B. Galpin2012

2. 1st Day

3. Dividing Out Technique

4. In Section 12.1 you learned how to use direct substitution to find the limit of certain functions. There are, however, situations that direct substitution fails. An example is If you did direct substitution, what would you get?

5. The above result has no meaning as a real number. It is called an indeterminate form because you cannot, from the form alone determine the limit. Make a table to estimate the above limit. What is your estimation? The limit is -5.

6. When the result of direct substitution is 0/0, then you know that the numerator and denominator have a common factor. This is when you use the dividing out technique and then direct substitution to find the limit.

8. The validity of the dividing out technique stems from the fact that if two functions agree at all but a single number c, they must have identical limit behavior. Since agree at all values of x except x = -3, g(x) can be used to find the limit of f(x).

9. Example 1

10. Find the limit: So you need to use the dividing out technique.

13. Rationalizing Technique

14. Another way to find the limits of some functions is first to rationalize the NUMERATOR of the function. This is called the rationalizing technique.

15. Example 2

16. Find the limit: Since you get the indeterminate form, you will use the rationalizing technique.

17. In this case you will rewrite the fraction by rationalizing the numerator.

18. Now use direct substitution.

19. End of the 1st Day Page 870 #1,3,5,8,9,12,13,15,17,19,21,31,33

21. 2nd Day

22. How can we look at a graph and decide if the limit at a certain value c exists? The graph does not go to a unique value L. This type of behavior can be described more concisely with the concept of a one-sided limit. Limit from the left Limit from the right

23. Example 1

24. Find the limit as x → 0 from the left and the limit as x → 0 from the right for

25. Existence of a Limit If f is a function and c and L are real numbers, then if and only if both the left and right limits EXIST and are EQUAL to L.

26. Example 2

27. Find the limit of f(x) as x approaches 2. Remember we do not care what happens to this function at x = 2 only as x approaches 2. Because of this we can just use direct substitution to find the left and right sided limits.

28. To find the left sided limit, that is when x < 2, we use x2 + 1. To find the right sided limit, that is when x > 2, we use

29. Since both one-sided limits exist and are equal to 5, then

30. Example 3

31. To ship a package overnight a delivery service chares $9 for the first pound and $1 for each additional pound or portion of a pound. Let x represent the weight of the package and let f(x) represent the shipping cost. Show that the limit of f(x) as x approaches 3 does not exist.

33. Since the one-sided limits are not equal, then the limit does not exist.

34. Throughout Pre-Calculus you have done problems using the difference quotient. The limit of the difference quotient is very important in Calculus and we will discuss it in Section 12.3. But first we will practice…

35. Example 4

36. For the function given by f(x) = 2x2 + 1, find

37. Decide what technique you want to use to find this limit.

39. Assignment: Page 870 #23,25,39,41,44,45,48,59,60. Quiz-tomorrow!!