Infinite Limits. Infinite Limits. Infinite Limits. Previously, we discussed the limits of rational functions with the indeterminate form 0/0.

Size: px

Start display at page:

Download "Infinite Limits. Infinite Limits. Infinite Limits. Previously, we discussed the limits of rational functions with the indeterminate form 0/0."

Grant Pitts
5 years ago
Views:

1 Infinite Limits Return to Table of Contents Infinite Limits Infinite Limits Previously, we discussed the limits of rational functions with the indeterminate form 0/0. Now we will consider rational functions where the denominator has a limit of zero, but the numerator does not. As a result, the function outgrows all positive or negative bounds. Infinite Limits 1

Infinite Limits We can define a limit like this as having a value of positive infinity or negative infinity: If the value of a function gets larger and larger without a bound, we say that the limit

2 Infinite Limits We can define a limit like this as having a value of positive infinity or negative infinity: If the value of a function gets larger and larger without a bound, we say that the limit has a value of positive infinity. If the value of a function gets smaller and smaller without a bound, we say that the limit has a value of negative infinity. Teacher Notes Next, we will use some familiar graphs to illustrate this situation. Infinite Limits Infinite Limits The figure to the right represents the function. Can we compute the limit? Infinite Limits 2

3 Left Hand Limit In this case, we should first discuss one sided limits. When x is approaching zero from the left the value of the function becomes smaller and smaller, so Infinite Limits Right Hand Limit When x is approaching zero from the right the value of function becomes larger and larger, so Infinite Limits 3

on the right represents the function. Can we compute the limit?

4 LHL RHL By definition of a limit, a two sided limit of this function does not exist, because the limit from the left and from the right are not the same: Infinite Limits Infinite Limits The figure on the right represents the function. Can we compute the limit? We see that the function outgrows all positive bounds as x approaches zero from the left and from the right, so we can say Infinite Limits 4

Thus, the two sided limit is: Infinite Limits Vertical Asymptote As we recall from algebra, the vertical lines near which the function grows without bound are vertical asymptotes.

5 Thus, the two sided limit is: Infinite Limits Vertical Asymptote As we recall from algebra, the vertical lines near which the function grows without bound are vertical asymptotes. Infinite limits give us an opportunity to state a proper definition of the vertical asymptote. Definition A line is called a vertical asymptote for the graph of a function if either: Infinite Limits 5

6 Example Find the vertical asymptote for the function. First, let us sketch a graph of this function. It is obvious from the graph, that. So, the equation of the vertical asymptote for this function is. Infinite Limits Number Line Method It seems that in the case when the denominator equals zero, but the numerator does not, as x approaches a certain number, we have to know what the graph looks like, before we can calculate a limit. Actually, this is not necessary. There is a number line method that will help us to solve these types of problems. Infinite Limits 6

7 Number Line Method If methods mentioned on previous pages are unsuccessful, you may need to use a number line to help you compute the limit. This is often helpful with one sided limits as well as limits involving absolute values, when you are not given a graph. 1. Make a number line marked with the value, "c" which x is approaching. 2. Plug in numbers to the right and left of "c" Remember... if the limits from the right and left do not match, the overall limit DNE. Infinite Limits Example 1 Find the limit: Step 1. Find all values of x that are zeros of the numerator and the denominator: Step 2. Draw a number line, plot these points. Infinite Limits 7

numbers inside the zeros interval, near 3, on both the left and right

8 Example 1 Step 3. Using the number line, test the sign of the value of the function at numbers inside the zeros interval, near 3, on both the left and right sides. (for example, pick and ). Infinite Limits Example 2 Use the number line method to find the limit: Infinite Limits 8

9 Find the limit: Infinite Limits Find the limit: Infinite Limits 9

10 Find the limit: Infinite Limits Find the limit: Infinite Limits 10

11 Find the limit: Infinite Limits Find the limit: Infinite Limits 11

12 Limits of Absolute Value and Piecewise Defined Functions Return to Table of Contents Special Limits Abs. Value & Piecewise Limits In the beginning of the unit we used the graphical approach to obtain the limits of the absolute value and the piecewise defined functions. However, we do not have to graph the given functions every time we want to compute a limit. Now we will offer algebraic methods to find limits of those functions. There is a reason why we discuss the absolute value and the piecewise functions in the same section: the graphs of these functions have two or more parts that are given by different equations. When you are trying to calculate a limit you have to be clear of which equation you have to use. Special Limits 12

Limits Involving Absolute Values We will start with the easiest

straightforward solution for this problem by simply

It would be identical solution for the similar problem: Special

examples were very straight forward, how do we approach the

: It is not possible here to substitute the value of x into the

But we can calculate one sided limits from the left and from

13 Limits Involving Absolute Values We will start with the easiest example of the absolute value function: There is a straightforward solution for this problem by simply substituting the value of x into the given formula. It would be identical solution for the similar problem: Special Limits Limits Involving Absolute Values While the previous examples were very straight forward, how do we approach the following situation?: It is not possible here to substitute the value of x into the given formula. But we can calculate one sided limits from the left and from the right. Any number that is bigger than 2 will turn the given expression into 1 and any number that is less than 2 will turn this expression into 1, so: Therefore, Special Limits 13

14 Example: Find the indicated limits: Special Limits Find the limit: Special Limits 14

15 Find the limit: Special Limits Find the limit: Special Limits 15

16 Find the limit: Special Limits Find the limit: Special Limits 16

17 Find the limit: Special Limits Piecewise Limits The key to calculating the limit of a piecewise function is to identify the interval that the x value belongs to. We can simply compute the limit of the function that is represented by the equation when x lies inside that interval. It may sound a lit bit tricky, however it is quite simple if you look at the next example. Special Limits 17

Example: Find the indicated limits of the piecewise function: Special Limits Two Sided Piecewise Limits A two sided limit of the piecewise function at the point where the formula changes is best

18 Example: Find the indicated limits of the piecewise function: Special Limits Two Sided Piecewise Limits A two sided limit of the piecewise function at the point where the formula changes is best obtained by first finding the one sided limits at this point. If limits from the left and the right equal the same number, the two sided limit exists, and is this number. If limits from the left and the right are not equal, than the two sided limit does not exist. Special Limits 18

19 Example: Find the indicated limit of the piecewise function: 1. First we will calculate the one sided limit from the left of the function represented by which formula? Special Limits Example: 2. Then we can calculate one sided limit from the right of the function represented by which formula? Special Limits 19

Example: Find the indicated limit of the piecewise function: 1. First, we will calculate one sided limit from the left of the function represented by which formula?

20 Example: Find the indicated limit of the piecewise function: 1. First, we will calculate one sided limit from the left of the function represented by which formula? Special Limits Example: Find the indicated limit of the piecewise function: 2. Then we can calculate one sided limit from the right of the function represented by which formula? Special Limits 20

21 Find the indicated limit of the given function: Special Limits Find the indicated limit of the given function: Special Limits 21

22 Find the indicated limit of the given function: Special Limits Find the indicated limit of the given function: Special Limits 22

23 Find the indicated limit of the given function: Special Limits Find the indicated limit of the given function: Special Limits 23

The indeterminate forms such as and others will lead us to the discussion of the end of the behavior function and

24 Limits of End Behavior Return to Table of Contents End Behaviors End Behavior In the previous sections we learned about an indeterminate form 0/0 and vertical asymptotes. The indeterminate forms such as and others will lead us to the discussion of the end of the behavior function and the horizontal asymptotes. The behavior of a function as x increases or decreases without bound (we write or ) is called the end behavior of the function. End Behaviors 24

End Behavior Let us recall the familiar function Can we compute the limits: End Behaviors End Behavior and Horizontal Asymptotes In general, we can use the following notation.

25 End Behavior Let us recall the familiar function Can we compute the limits: End Behaviors End Behavior and Horizontal Asymptotes In general, we can use the following notation. If the value of a function eventually get as close as possible to a number L as x increases without bound, then we write: Similarly, if the value of a function eventually get as close as possible to a number L as x decreases without bound, then we write: We call these limits Limits at Infinity and the line is the horizontal asymptote of the function. End Behaviors 25

26 End Behavior The figure below illustrates the end behavior of a function and the horizontal asymptote. y=1 1 End Behaviors Example: Figure on the right is the graph of the function. As you can see from the graph, so, this graph has two horizontal asymptotes: and End Behaviors 26

27 Example: The figure on the right is the graph of the function As you can see from the graph, so, this graph has only one horizontal asymptote which is the line. End Behaviors Different Scenarios When a limit goes to infinity we are looking for any horizontal asymptotes and a number of things can happen. Obviously, the first example isn't a problem. But what about the second and third examples? Not all s are the same. End Behaviors 27

28 Infinite Limits Consider Is the limit 1? It is not, because if we reduce the rational expression before substituting, we will get the limit : In calculus, we have to divide each term by the highest power of x, then take the limit. Remember that a number/ is 0 as we have seen in a beginning of the section. For example, End Behaviors Find: Example 1 End Behaviors 28

Example 2 Find: End Behaviors Rule 1 There are

$infinity: First, write f (x) as a fraction.$ Rule 1: If the highest power of x appears in

29 Example 2 Find: End Behaviors Rule 1 There are three rules that help you to solve problems when x is approaching positive or negative infinity: First, write f (x) as a fraction. Rule 1: If the highest power of x appears in the denominator (bottom heavy), then Examples: End Behaviors 29

negative numbers. Look closely at the examples on next pages to understand this rule clearly.

30 Rule 2 Rule 2. If the highest power of x appears in the numerator (top heavy), then In order to determine the resulting sign of the infinity you will need to plug in very large positive or very large negative numbers. Look closely at the examples on next pages to understand this rule clearly. End Behaviors Examples For rational functions, the end behavior matches the end behavior of the quotient of the highest degree term in the numerator divided by the highest degree term in the denominator. So, in this case the function will behave as. When x approaches positive infinity, the limit of the function will go to positive infinity. When x approaches negative infinity in the same problem, the limit of function will go to negative infinity. End Behaviors 30

function will go to negative infinity. End Behaviors Rule 3 Rule 3.

31 Examples The end behavior of this function matches the end behavior of the polynomial. When x is approaching positive infinity or negative infinity, the limit of the function will go to negative infinity. End Behaviors Rule 3 Rule 3. If the highest power of x appears both in the numerator and denominator (powers equal), then End Behaviors 31

32 Examples End Behaviors Try These on Your Own End Behaviors 32

Limits Involving Radicals You have to be very careful with limits involving radicals.

But be careful before dividing the coefficients. reduces to 2. But the limit is going to.

End Behaviors Limits Involving Radicals There is a more precise way to solve all problems like this.

33 Limits Involving Radicals You have to be very careful with limits involving radicals. There is a problem: By given rules, we check on the highest powers: they are the same. But be careful before dividing the coefficients. reduces to 2. But the limit is going to. Since the denominator is always positive, only the numerator is negative. End Behaviors Limits Involving Radicals There is a more precise way to solve all problems like this. We can divide the numerator and the denominator by the highest power, which is, and using the fact that as the value of x under consideration is negative, so we can replace by x. End Behaviors 33

34 20 Find the indicated limit. End Behaviors 21 Find the indicated limit. End Behaviors 34

35 22 A B C D End Behaviors 23 A B C D End Behaviors 35

36 24 A B C D End Behaviors Which is getting bigger faster? Why? End Behaviors 36

$fraction, then solve like /.$

37 Using Conjugates Use conjugates to rewrite the expression as a fraction, then solve like /. End Behaviors Example Evaluate: End Behaviors 37

38 End Behaviors Trig Limits Return to Table of Contents Trig Limits 38

39 3 Trig Limits to Know When evaluating limits, if you can use substitution and get a number, great! But what if you get an indeterminate form? Trig Limits Examples Trig Limits 39

40 Examples Trig Limits Find the limit of Trig Limits 40

41 Find the limit of Trig Limits Find the limit of Trig Limits 41

42 Find the limit of Trig Limits Find the limit of Trig Limits 42

Squeeze (or Sandwich) Theorem These limits were found by using

This theorem states that if you want to find the limit of a

that is always greater,, such that Trig Limits Limits Summary &

the possible strategies used for limit questions.

43 Squeeze (or Sandwich) Theorem These limits were found by using the Squeeze (or Sandwich) Theorem. This theorem states that if you want to find the limit of a function,, pick a function,, which is always less than and one that is always greater,, such that Trig Limits Limits Summary & Plan of Attack It can seem very overwhelming to think about all the possible strategies used for limit questions. Hopefully, the next pages will provide you with a game plan to approach limit problems with confidence. Trig Limits 43

Limits with Graphs Questions involving limits and graphs will typically ask one of the following: 1.

Follow/ Trace the graph towards the given x value "a" from the right and your answer is the y value as you reach "a". 3.

If not, the limit DNE. 4. Find the y value that matches up with the given x value "a".

44 Limits with Graphs Questions involving limits and graphs will typically ask one of the following: 1. Follow/ Trace the graph towards the given x value "a" from the left and your answer is the y value as you reach "a". 2. Follow/ Trace the graph towards the given x value "a" from the right and your answer is the y value as you reach "a". 3. Follow/ Trace the graph towards the given x value "a" from BOTH sides. If you end up at the same y value, this is your answer. If not, the limit DNE. 4. Find the y value that matches up with the given x value "a". In other words, where is the graph "filled in " for the given x value. 5. If the question asks for continuity, make sure that the limit exists, the function exists, and that they are equal to each other! Trig Limits Limits with Graphs Teacher Notes Trig Limits 44

$!! If you get try one of the following: Teacher Notes the limit DNE. Try to factor the top and/ or bottom of the fraction to cancel pieces. Then substitute into the simplified expression!$ If you see a multiply by the conjugate, simplify and then substitute into the simplified expression!

If you see a multiply by the conjugate, simplify and then substitute into the simplified expression!

45 If you get a real number out, you're finished! If you get a Limits without Graphs Always try to substitute the value into your expression FIRST!!! If you get try one of the following: Teacher Notes the limit DNE. Try to factor the top and/ or bottom of the fraction to cancel pieces. Then substitute into the simplified expression! If you see a multiply by the conjugate, simplify and then substitute into the simplified expression! Recognize special Trig Limits to simplify, then substitute into the simplified expression, if needed! *if none of these are possible, try the Number Line Method Trig Limits Limits Approaching higher degree on bottom higher degree on top equivalent degrees (rule 1) (rule 2) coefficients of highest degrees (rule 3) Teacher Notes Trig Limits 45

46 Continuity Return to Table of Contents Continuity Continuity At what points do you think the graph below is continuous? At what points do you think the graph below is discontinuous? a b c d e f g h What should the definition of continuous be? Continuity 46

47 AP Calculus Definition of Continuous 1) exists 2) exists 3) This definition shows continuity at a point on the interior of a function. For a function to be continuous, every point in its domain must be continuous. Continuity Continuity at an Endpoint Replace step 3 in the previous definition with: Left Endpoint: Right Endpoint: Continuity 47

48 Types of Discontinuity Infinite Jump Removable Essential Continuity 25 Given the function decide if it is continuous or not. If it is not state the reason it is not. A continuous B f(a) does does not not exist exist for all afor all a C does not exist for all a D is not true for all a Continuity 48

49 26 Given the function decide if it is continuous or not. If it is not state the reason it is not. A continuous B does not exist for all a C does not exist for all a D is not true for all a Continuity 27 Given the function decide if it i s continuous or not. If it is not state the reason it is not. A continuous B does not exist for all a C D does not exist for all a and is not true for all a does not ex Continuity 49

true for all a Continuity Making a Function Continuous Removable discontinuities come from rational

50 28 Given the function decide if it is continuous or not. If it is not state the reason it is not. A continuous B f(a) does does not not exist exist for all afor all a C D does not exist for all a is not true for all a Continuity Making a Function Continuous Removable discontinuities come from rational functions. A piecewise function can be used to fill the "hole". What should k be so that continuous? is Continuity 50

51 29 What value(s) would remove the discontinuity(s)of the given function? A 3 B 2 C 1 D 1/2 F 1/2 G 1 H 2 I 3 E 0 J DNE Continuity 30 What value(s) would remove the discontinuity(s) of the given function? A 3 B 2 C 1 D 1/2 F 1/2 G 1 H 2 I 3 E 0 J DNE Continuity 51

52 31 What value(s) would remove the discontinuity(s) of the given function? A 3 B 2 C 1 D 1/2 F 1/2 G 1 H 2 I 3 E 0 J DNE Continuity 32 What value(s) would remove the discontinuity(s) of the given function? A 3 B 2 C 1 D 1/2 E 0 F 1/2 G 1 H 2 I 3 J DNE Continuity 52

53 Making a Function Continuous, find a so that is continuous. Both 'halves' of the function are continuous. The concern is making Continuity What value of k will make the function continuous? Continuity 53

54 What value of k will make the function continuous? Continuity Intermediate Value Theorem Return to Table of Contents IVT 54

55 The Intermediate Value Theorem The characteristics of a function on closed continuous interval is called The Intermediate Value Theorem. If is a continuous function on a closed interval [a,b], then takes on every value between and. f(b) f(a) a b This comes in handy when looking for zeros. IVT Finding Zeros Can you use the Intermediate Value Theorem to find the zeros of this function? X Y IVT 55

56 33 Give the letter that lies in the same interval as a zero of this continuous function. A B C D X Y IVT 34 Give the letter that lies in the same interval as a zero of this continuous function. A B C D X Y IVT 56

57 35 Give the letter that lies in thesame interval as a zero of this continuous function. A B C D X Y IVT Difference Quotient Return to Table of Contents Difference Quotient 57

Difference Quotient Now that you are familiar with the different types of limits, we can discuss real life applications

You definitely noticed that in all formulas stated in the previous section, the numerator is presented as a difference

Such as: We call this a Difference Quotient or an Average Rate of Change.

58 Difference Quotient Now that you are familiar with the different types of limits, we can discuss real life applications of this very important mathematical term. You definitely noticed that in all formulas stated in the previous section, the numerator is presented as a difference of a function or, in another words, as a change of a function; and the denominator is a point difference. Such as: We call this a Difference Quotient or an Average Rate of Change. If we consider a situation when approaches ( ), which means, or then it then describes the Instantaneous Rate of Change. Difference Quotient Example Draw a possible graph of traveling 100 miles in 2 hours. 100 Distance (miles) 1 Time (hrs) 2 Difference Quotient 58

59 Average Rate of Change Using the graph on the previous slide: What is the average rate of change for the trip? Is this constant for the entire trip? What formula could be used to find the average rate of change between 45 minutes and 1 hour? Difference Quotient Average Rate of Change The slope formula of represents the Velocity or Average Rate of Change. This is the slope of the secant line from to. Suppose we were looking for Instantaneous Velocity at 45 minutes, what values of and should be used? Is there a better approximation? Difference Quotient 59

The Difference Quotient The closer and get to one another the better the approximation is.

and The slope between them would be And since we want h to "disappear" we use Difference

velocity, which is the slope of the tangent line at a point.

60 The Difference Quotient The closer and get to one another the better the approximation is. Let h represent a very small value so that are 2 points that are very close to each other. and The slope between them would be And since we want h to "disappear" we use Difference Quotient Derivative The limit of the Difference Quotient when h >0 gives the instantaneous velocity, which is the slope of the tangent line at a point. A derivative is used to find the slope of a tangent line. So, the Difference Quotient can be used to find a derivative algebraically. Difference Quotient 60

61 Example of the Difference Quotient Find the slope of the tangent line to the function at. Difference Quotient Example of the Difference Quotient Find an equation that can be used to find the slope of the tangent line at any point on the function Difference Quotient 61

62 36 What is the slope of the tangent line of at? Difference Quotient 37 What is the slope of the tangent line of at? Difference Quotient 62

63 38 What expression can be used to find the slope of any tangent line to? A B C D Difference Quotient 39 What expression can be used to find the slope of any tangent line to? A B C D Difference Quotient 63

64 40 Find the slope of the tangent at Difference Quotient 41 Find the slope of the tangent at Difference Quotient 64

65 42 Find the slope of the tangent at Difference Quotient 65

AP Calculus AB. Limits & Continuity.

AP Calculus AB. Limits & Continuity. 1 AP Calculus AB Limits & Continuity 2015 10 20 www.njctl.org 2 Table of Contents click on the topic to go to that section Introduction The Tangent Line Problem Definition of a Limit and Graphical Approach