1. 3-2 Limits Limit as x approaches a number

2. Just what is a limit? A limit is what the ___________________________ __________________________________________ We may not actually get there. BUT a limit is not what you actually get to, but appear to get to. Asymptote: _________________________

3. Lets look at a couple Graphs

4. What is the way to solve it? The easiest way to find the limit value is to plug the number in. Find the following

5. What about What is the restriction? (what can’t the denominator be) Factor the top and see if any terms cancel out If a term cancels out _______________________ ________________________________________ Graph the above equation

6. What if…. What do you think the answer is if you plug in the number and get ? What do you think the answer is if you plug in the number and get ?

7. The Harmonic Sequence The process used to find limits as x  is based on the Harmonic Sequence The is 0. Think about it. What about As x gets really really huge, what will the reciprocal of the fxns approach?

8. Some rules of limits The great thing about limits is that the limit of something complicated can be done as the limit of all the pieces.

9. Taking the limit of Equations Steps to Solve: Divide each term by the highest overall power you see in the problem. Then evaluate each of the pieces. Then take the limit of each term.

10. Example

11. Group Problems Find the following Now there is a shortcut “trick” to these problems. WITHOUT TALKING TO ANYONE tonight see if you can figure it out.

12. 3-2 Limits Day 2

13. Anyone figure out the short cut? That’s right. You look for the overall high power. Overall High Power in top = _______________ Overall High Power in bottom = ____________ High Power in top and bottom are the same =_________________________________ __________________________________

14. Examples of Graphs So, you can see that the graphs have these vertical and horizontal lines that act as boundary lines. These are called Asymptotes. ______________________________________ ______________________________________ ______________________________________

15. Asymptotes Vertical Asymptotes (VA): ___________________ ________________________________________ Hole: ___________________________________ Why does it still count if it goes away? ________________________________________ Horizontal Asymptotes (HA): ________________ ________________________________________

16. Step 1 – Find Vertical and Horizontal Asymptotes

17. Step 2 - Plot 3 points on each side of the vertical asymptote(s). Graph

18. 4 **Graphs can cross a Horizontal Asym but not a vert Asym.

19. 3-3 Oblique Asymptotes

20. OK – to review for just a minute If a VA cancels out, ___________________ ___________________________________ If a VA doesn’t cancel out, ______________ ___________________________________ FYI: _______________________________ ___________________________________

21. So what is an oblique? Did you notice that all of the graphs that had vertical asymptotes also had limits? That is, the only functions that didn’t have limits had holes. What if you have no limit to the function, but as well have a vertical asymptote? Such as, Lets go graphing!graphing

23. What happened? Well the vertical asymptote stayed, but the graph didn’t level. There was a diagonal line that acted as a boundary line. This diagonal line is the __________________ ___________________________________ So, lets figure out how to find the OA.

24. Finding OB Asymptote When the limit does not exist and there is a restriction, _________________________ ___________________________________ To find the OA, ___________________ into the numerator and ignore the remainder. That ______ is the oblique asymptote. Then graph the function the same way as if there was a VA.

25. Cancels Take the Limit Check denominator of F(x) Plot points and graph the function There will be a hole Exists Plot 3 points on each side of VA Divide Denominator into Numerator DNE Doesn’t Plot the OA as a dashed line; then plot 3 points on either side” of the OA

26. Graph the following 2

27. 2

28. Group Problem 1 1

29. 3.4 Day 1 Solving Fractional Equations

30. 1.Find the x intercept of the graph X-int? ________________________

31. Practice solving Equations

32. Isolating Variables:

33. Solving the Inequalities ______________________________ _______________________________ ______________________________

34. Example

35. 3-4 Day 2 Word Problems

36. Work Rate Problems ____________________________________ _____________________________________ ___________________________________ ____________________________________

37. Jan can tile a floor in 14 hours. Together, Jan and her helper can tile the same floor together in 9 hours. How long would it take Bill to do the job alone? Work Rate x Time = Work done

38. 1.The denominator of a fraction is 1 less than twice the numerator. If 7 is added to both numerator and denominator, the resulting fraction has a value of 7/10. Find the original fraction. Examples

39. Example A student received grades of 72, 75 and 78 on three tests. What must he score on the next test to average a 80?

40. 3.6 Synthetic Division

41. What is Synthetic Division? Synthetic Division ____________________ ___________________________________ 1.________________________________ 2.________________________________ 3.________________________________ ___________________________________

42. Lets try a problem Please divide by long division.

43. This is Synthetic Division This is the equivalent problem in synthetic division form: ___________________________________

44. This is synthetic Division Try synthetic Division and see what you get: 1 4 3 1 4 3 -3

45. Here’s another problem with a bit of a twist. If your last name begins with A-M, do this problem by long division. If your last name begins with N-Z, do this problem by synthetic division.

46. What did you notice? Answer is doubled. When there is a number in front of the x, __________________________________ __________________________________ __________________________________ __________________________________ **One other rule If one of the x’s are missing plug a zero in its place!

47. 3.6 Day 2 Why Synthetic Division? What use is this method, besides the obvious saving of time and paper?

48. The Remainder Theorem If is not a factor of F(x), then ___________________________________ That is

49. The Factor Theorem If is a factor of F(x) then _________ When we talk about roots, it’s the same as zeros. Set equal to zero and solve.

50. How does this apply? 1.Find F(2) if 2.Is (x – 2) a factor of

51. Factor Completely and find the roots:

52. Find the polynomial that has as roots 1, -1 and 7 Polynomial = ________________

53. 3.7 Solving Polynomial Equations That is, finding all the roots of P(x) without a head start

54. There are 5 main rules we will use to determine possible rational roots. There are others that you can read about in the book, but these 5 are the basic ones you narrow down the possibilities. Remember: when you divide synthetically, if the remainder = 0 _______________________. If the remainder ≠0 then the number is not a root and never will be.

55. Rule #1 The only possible real rational roots are Where

56. Rule #2 If the signs of all the terms in the polynomial are +, ____________________________ ___________________________________ Think about this, using

57. Rule #3 If the signs of the terms of the polynomial alternate 1 to 1 (that is + – + – + –) then __________________________________ ___________________________________ If a term is missing, it is ok to assign it a + or – value to make it fit this rule.

58. Rule #4 If you add all the coefficients and get 0, then 1 is a root. Otherwise 1 is not a root (and never will be a root ever). This is a good one. Essentially if it works, you have your start point.

59. Rule #5 Change the signs of the odd powered coefficients and then add. If you get 0, then -1 is a root. Otherwise, -1 is not a root. Sometimes this one isn’t worth the effort.

60. Why these rules? You will have a list of possibilities (and maybe a definite) with which to start synthetically dividing. Remember – the goal of the problem is to find all zeroes (or factor). A zero is something whose factor divides evenly into a function. Therefore, synthetically you want to get a remainder of 0.