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LIMITS - Introduction

LIMITS - Introduction. Many problems in algebra and trig look at the behavior of functions at given values. A limit however looks at the functions behavior NEAR a specific value. LIMITS - Introduction.

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LIMITS - Introduction

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  1. LIMITS - Introduction Many problems in algebra and trig look at the behavior of functions at given values. A limit however looks at the functions behavior NEAR a specific value.

  2. LIMITS - Introduction Many problems in algebra and trig look at the behavior of functions at given values. A limit however looks at the functions behavior NEAR a specific value.

  3. LIMITS - Introduction Many problems in algebra and trig look at the behavior of functions at given values. A limit however looks at the functions behavior NEAR a specific value. I am going to use a table of values where “x” gets close to 2 but never reaches 2

  4. LIMITS - Introduction Many problems in algebra and trig look at the behavior of functions at given values. A limit however looks at the functions behavior NEAR a specific value. I am going to use a table of values where “x” gets close to 2 but never reaches 2

  5. LIMITS - Introduction Many problems in algebra and trig look at the behavior of functions at given values. A limit however looks at the functions behavior NEAR a specific value. I am going to use a table of values where “x” gets close to 2 but never reaches 2

  6. LIMITS - Introduction Many problems in algebra and trig look at the behavior of functions at given values. A limit however looks at the functions behavior NEAR a specific value. As you can see, as I get really close to 2 , the function gets close to 5.

  7. LIMITS - Introduction Many problems in algebra and trig look at the behavior of functions at given values. A limit however looks at the functions behavior NEAR a specific value. In this table, I got really close to 2 from the other side.

  8. LIMITS - Introduction Many problems in algebra and trig look at the behavior of functions at given values. A limit however looks at the functions behavior NEAR a specific value. In this table, I got really close to 2 from the other side. Once again, the function got very close to 5.

  9. LIMITS - Introduction

  10. LIMITS - Introduction

  11. LIMITS - Introduction Method for evaluating limits: 1. Does “c” create undefined ? If yes, you must go to steps two or three ** if no undefined is created, just plug “c” into the function 2. Try to factor and get rid of the undefined. Then plug “c” into the function 3. Use a table of values getting close to “c” from the left and right. If the values are approaching THE SAME result, the limit exists.

  12. LIMITS - Introduction Method for evaluating limits: 1. Does “c” create undefined ? If yes, you must go to steps two or three ** if no undefined is created, just plug “c” into the function 2. Try to factor and get rid of the undefined. Then plug “c” into the function 3. Use a table of values getting close to “c” from the left and right. If the values are approaching THE SAME result, the limit exists. One final note: if you use a table, to show the value of the limit from the left or right of “c”, we will use a superscript “+” or “-” sign to show the direction…

  13. LIMITS - Introduction Method for evaluating limits: 1. Does “c” create undefined ? If yes, you must go to steps two or three ** if no undefined is created, just plug “c” into the function 2. Try to factor and get rid of the undefined. Then plug “c” into the function 3. Use a table of values getting close to “c” from the left and right. If the values are approaching THE SAME result, the limit exists. One final note: if you use a table, to show the value of the limit from the left or right of “c”, we will use a superscript “+” or “-” sign to show the direction… from the right from the left

  14. LIMITS - Introduction Method for evaluating limits: 1. Does “c” create undefined ? If yes, you must go to steps two or three ** if no undefined is created, just plug “c” into the function 2. Try to factor and get rid of the undefined. Then plug “c” into the function 3. Use a table of values getting close to “c” from the left and right. If the values are approaching THE SAME result, the limit exists.

  15. LIMITS - Introduction Method for evaluating limits: 1. Does “c” create undefined ? If yes, you must go to steps two or three ** if no undefined is created, just plug “c” into the function 2. Try to factor and get rid of the undefined. Then plug “c” into the function 3. Use a table of values getting close to “c” from the left and right. If the values are approaching THE SAME result, the limit exists. c = 4 will not create an undefined situation, so just find f(4)

  16. LIMITS - Introduction Method for evaluating limits: 1. Does “c” create undefined ? If yes, you must go to steps two or three ** if no undefined is created, just plug “c” into the function 2. Try to factor and get rid of the undefined. Then plug “c” into the function 3. Use a table of values getting close to “c” from the left and right. If the values are approaching THE SAME result, the limit exists. c = 4 will not create an undefined situation, so just find f(4)

  17. LIMITS - Introduction Method for evaluating limits: 1. Does “c” create undefined ? If yes, you must go to steps two or three ** if no undefined is created, just plug “c” into the function 2. Try to factor and get rid of the undefined. Then plug “c” into the function 3. Use a table of values getting close to “c” from the left and right. If the values are approaching THE SAME result, the limit exists.

  18. LIMITS - Introduction Method for evaluating limits: 1. Does “c” create undefined ? If yes, you must go to steps two or three ** if no undefined is created, just plug “c” into the function 2. Try to factor and get rid of the undefined. Then plug “c” into the function 3. Use a table of values getting close to “c” from the left and right. If the values are approaching THE SAME result, the limit exists. c = -3 will create an undefined situation, so now try to factor

  19. LIMITS - Introduction Method for evaluating limits: 1. Does “c” create undefined ? If yes, you must go to steps two or three ** if no undefined is created, just plug “c” into the function 2. Try to factor and get rid of the undefined. Then plug “c” into the function 3. Use a table of values getting close to “c” from the left and right. If the values are approaching THE SAME result, the limit exists. c = -3 will create an undefined situation, so now try to factor

  20. LIMITS - Introduction Method for evaluating limits: 1. Does “c” create undefined ? If yes, you must go to steps two or three ** if no undefined is created, just plug “c” into the function 2. Try to factor and get rid of the undefined. Then plug “c” into the function 3. Use a table of values getting close to “c” from the left and right. If the values are approaching THE SAME result, the limit exists. c = -3 will create an undefined situation, so now try to factor

  21. LIMITS - Introduction Method for evaluating limits: 1. Does “c” create undefined ? If yes, you must go to steps two or three ** if no undefined is created, just plug “c” into the function 2. Try to factor and get rid of the undefined. Then plug “c” into the function 3. Use a table of values getting close to “c” from the left and right. If the values are approaching THE SAME result, the limit exists.

  22. LIMITS - Introduction Method for evaluating limits: 1. Does “c” create undefined ? If yes, you must go to steps two or three ** if no undefined is created, just plug “c” into the function 2. Try to factor and get rid of the undefined. Then plug “c” into the function 3. Use a table of values getting close to “c” from the left and right. If the values are approaching THE SAME result, the limit exists. c = 1 will create an undefined situation, so now try to factor

  23. LIMITS - Introduction Method for evaluating limits: 1. Does “c” create undefined ? If yes, you must go to steps two or three ** if no undefined is created, just plug “c” into the function 2. Try to factor and get rid of the undefined. Then plug “c” into the function 3. Use a table of values getting close to “c” from the left and right. If the values are approaching THE SAME result, the limit exists. c = 1 will create an undefined situation, so now try to factor

  24. LIMITS - Introduction Method for evaluating limits: 1. Does “c” create undefined ? If yes, you must go to steps two or three ** if no undefined is created, just plug “c” into the function 2. Try to factor and get rid of the undefined. Then plug “c” into the function 3. Use a table of values getting close to “c” from the left and right. If the values are approaching THE SAME result, the limit exists. c = 1 will create an undefined situation, so now try to factor OK, so we can’t get our (x-1) to factor out, so we need to use a table of values.

  25. LIMITS - Introduction Method for evaluating limits: 1. Does “c” create undefined ? If yes, you must go to steps two or three ** if no undefined is created, just plug “c” into the function 2. Try to factor and get rid of the undefined. Then plug “c” into the function 3. Use a table of values getting close to “c” from the left and right. If the values are approaching THE SAME result, the limit exists. TABLE OF VALUES

  26. LIMITS - Introduction Method for evaluating limits: 1. Does “c” create undefined ? If yes, you must go to steps two or three ** if no undefined is created, just plug “c” into the function 2. Try to factor and get rid of the undefined. Then plug “c” into the function 3. Use a table of values getting close to “c” from the left and right. If the values are approaching THE SAME result, the limit exists. TABLE OF VALUES As you can see, our functions value is getting very large…approaching infinity as we get closer from the left of 1.

  27. LIMITS - Introduction Method for evaluating limits: 1. Does “c” create undefined ? If yes, you must go to steps two or three ** if no undefined is created, just plug “c” into the function 2. Try to factor and get rid of the undefined. Then plug “c” into the function 3. Use a table of values getting close to “c” from the left and right. If the values are approaching THE SAME result, the limit exists. TABLE OF VALUES As you can see, our functions value is getting very large…approaching infinity as we get closer from the left of 1.

  28. LIMITS - Introduction Method for evaluating limits: 1. Does “c” create undefined ? If yes, you must go to steps two or three ** if no undefined is created, just plug “c” into the function 2. Try to factor and get rid of the undefined. Then plug “c” into the function 3. Use a table of values getting close to “c” from the left and right. If the values are approaching THE SAME result, the limit exists. TABLE OF VALUES As we come in from the right of one, the function values are approaching negative infinity…

  29. LIMITS - Introduction Method for evaluating limits: 1. Does “c” create undefined ? If yes, you must go to steps two or three ** if no undefined is created, just plug “c” into the function 2. Try to factor and get rid of the undefined. Then plug “c” into the function 3. Use a table of values getting close to “c” from the left and right. If the values are approaching THE SAME result, the limit exists. TABLE OF VALUES As we come in from the right of one, the function values are approaching negative infinity…

  30. LIMITS - Introduction Method for evaluating limits: 1. Does “c” create undefined ? If yes, you must go to steps two or three ** if no undefined is created, just plug “c” into the function 2. Try to factor and get rid of the undefined. Then plug “c” into the function 3. Use a table of values getting close to “c” from the left and right. If the values are approaching THE SAME result, the limit exists.

  31. LIMITS - Introduction Limits at infinity

  32. LIMITS - Introduction Limits at infinity

  33. LIMITS - Introduction Limits at infinity It is difficult to test infinity into a function since it is not a real number, just a notation to show something will increase or decrease without bound as “x” approaches “c”.

  34. LIMITS - Introduction Limits at infinity It is difficult to test infinity into a function since it is not a real number, just a notation to show something will increase or decrease without bound as “x” approaches “c”. So our method for finding limits at infinity will differ from our previous examples…

  35. LIMITS - Introduction Limits at infinity

  36. LIMITS - Introduction Limits at infinity This makes sense. If I keep replacing “x” with larger and larger values, the result will get smaller and smaller.

  37. LIMITS - Introduction Limits at infinity This makes sense. If I keep replacing “x” with larger and larger values, the result will get smaller and smaller. This principle is called the “Big – Little” principle. We will utilize it to solve limits at infinity.

  38. LIMITS - Introduction Limits at infinity

  39. LIMITS - Introduction Limits at infinity - Divide both numerator and denominator by the highest exponent of “x” present in the function

  40. LIMITS - Introduction Limits at infinity - After simplifying, you can see you have the rule above

  41. LIMITS - Introduction Limits at infinity These values will both = 0 - After simplifying, you can see you have the rule above

  42. LIMITS - Introduction Limits at infinity These values will both = 0

  43. LIMITS - Introduction Limits at infinity

  44. LIMITS - Introduction Limits at infinity - Divide both numerator and denominator by the highest exponent of “x” present in the function

  45. LIMITS - Introduction Limits at infinity - again, after simplifying, you can see you have the rule above

  46. LIMITS - Introduction Limits at infinity - Replace the small values with 0

  47. LIMITS - Introduction Limits at infinity

  48. LIMITS - Introduction Limits at infinity

  49. LIMITS - Introduction Limits at infinity Replace small values with zero…

  50. LIMITS - Introduction Unbounded Limits

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