Ch 3.1 Limits Solve limits algebraically using direct substitution and by factoring/cancelling Solve limits graphically – including the wall method Solve limits using Derive/Maple

Slide 1-2 THE LIMIT (L) OF A FUNCTION IS THE VALUE THE FUNCTION (y) APPROACHES AS THE VALUE OF (x) APPROACHES A GIVEN VALUE.

Slide 1-3 Direct Substituion Easiest way to solve a limit Can’t use if it gives an undefined answer

Slide 1-4 Table method and direct substitution method. because as x gets closer and closer to 2, x cubed gets closer and closer to 8. (using table on next slide)

Slide 1-5 Can use a table to find a limit. X y This is just for explanation purposes. We won’t use this method.

Slide 1-6 Graphically y =

Slide 1-7 In the previous example it was fairly evident that the limit was 8, because when we replaced x with 2 (using direct substitution) the function had a value of 8. This is not always as evident. Find the limit below.

Slide 1-8 Rewrite before substituting Factor and cancel common factors – then do direct substitution. The answer is 4.

Slide 1-9 means x approaches a from the right means x approaches a from the left

Slide 1-10 When finding the limit of a function it is important to let x approach a from both the right and left. If the same value of L is approached by the function then the limit exist and

Slide 1-11 THEOREM: As x approaches a, the limit of f (x) is L, if the limit from the left exists and the limit from the right exists and both limits are L. That is, If 1) and 2) Then

Slide 1-12 Graphs can be used to determine the limit of a function. Find the following limits.

Slide 1-13 Limit Graphically example 1 Find: (DNE) 1.1 Limits: A Numerical and Graphical Approach

Slide 1-14 (DNE) Therefore, Example 1 - Answer

Slide 1-15 The “Wall” Method: As an alternative approach to Example 1, we can draw a “wall” at x = 1, as shown in blue on the following graphs. We then follow the curve from left to right with pencil until we hit the wall and mark the location with an ×, assuming it can be determined. Then we follow the curve from right to left until we hit the wall and mark that location with an ×. If the locations are the same, we have a limit. Otherwise, the limit does not exist.

Slide 1-16 Thus for Example 1: now try this one Find 1.1 Limits: A Numerical and Graphical Approach

Slide 1-17 a) Limit Graphically Observe on the graph that: 1) and 2) Therefore, 1.1 Limits: A Numerical and Graphical Approach

Slide 1-18 We can also use Derive to evaluate limits.

Slide 1-19 Use Derive to find the indicated limit.

Slide 1-20 Limits at infinity Sometimes we will be concerned with the value of a function as the value of x increases without bound. These cases are referred to as limits at infinity and are denoted:

Slide 1-21 For polynomial functions the limit will be + or – infinity. Use the leading term to identify the end behavior. Use it to determine whether the answer is + or – infinity.

Slide 1-22 For rational functions, the limit at infinity is the same as the horizontal asymptote (y = L) of the function. Recall the method of finding the horizontal asymptote depends on the degrees of the numerator and denominator

Slide 1-23 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Limit Graphically Observe on the graph that, again, you can only approach ∞ from the left. Therefore, Answer : 3

Slide 1-24 Find each limit

Slide 1-25 Answers a) 5 b) 100 c) No limit exists, but graph to see if Answer to c is

Slide 1-26 Infinite limits When considering f(x) may increase or decrease without bound (becomes infinite) as x approaches a. In these cases the limit is infinite and

Slide 1-27 When this occurs the line x = a is a vertical asymptote. Polynomial functions do not have vertical asymptotes, but rational functions may have vertical asymptotes.

Slide 1-28 b) Limit Graphically Observe on the graph that: 1) and 2) Therefore, does not exist.

Slide 1-29 We will use graphs to find the answers to these. Graph each on the computer to determine the answer.

Slide 1-30 Answers a) DNE b) c)

Slide 1-31 The cost (in dollars) for manufacturing a particular videotape is where x is the number of tapes produced. The average cost per tape, denoted by is found by dividing C(x) by x. Find and interpret

Slide 1-32 Answers As the number of tapes increases, the average cost per tape approaches $6.