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3-2 Limits

3-2 Limits. Limit as x approaches a number. Just what is a limit?. A limit is what the ___________________________ __________________________________________ __________________________________________ __________________________________________

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3-2 Limits

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  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 -1 **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!

  22. 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.

  23. 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.

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

  25. Graph the following 2

  26. Graph the following 2

  27. Group Problem 1 1

  28. 3.4 Day 1 Solving Fractional Equations

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

  30. Practice solving Equations

  31. Isolating Variables:

  32. Solving the Inequalities • ______________________________ • _______________________________ • ______________________________ • ______________________________

  33. Example

  34. 3-4 Day 2 Word Problems

  35. Work Rate Problems • ____________________________________ • _____________________________________ • _____________________________________ • _____________________________________ • ___________________________________ • ____________________________________

  36. 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

  37. Examples • 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.

  38. 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?

  39. 3.6 Synthetic Division

  40. What is Synthetic Division? Synthetic Division ____________________ ___________________________________ • ________________________________ • ________________________________ • ________________________________ ___________________________________ ___________________________________

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

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

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

  44. 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.

  45. 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!

  46. 3.6 Day 2Why Synthetic Division? What use is this method, besides the obvious saving of time and paper?

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

  48. 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.

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

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