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Pre Cal 122 lesson with notes 5th.notebook 1 May 09, 2017 Dec 128:02 PM 122 Evaluating Limits Algebraically Daily Outcomes: I can evaluate limits of polynomial and rational functions at selected points I can evaluate limits of polynomial and rational functions at infinity What limit does x approach when the light is at its minimum? maximum? May 911:36 AM

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Page 1: Pre Cal 12-2 lesson with notes 5th.notebookimages.pcmac.org/.../Uploads/Forms/Pre_Cal_Lesson_12-2.pdf · 2019. 9. 25. · Pre Cal 122 lesson with notes 5th.notebook 1 May 09, 2017

Pre Cal 12­2 lesson with notes 5th.notebook

1

May 09, 2017

Dec 12­8:02 PM

12­2 Evaluating Limits Algebraically

Daily Outcomes:I can evaluate limits of polynomial and rational functions at 

selected pointsI can evaluate limits of polynomial and rational functions at 

infinity

What limit does x approach when the light is at its minimum? maximum?

May 9­11:36 AM

Page 2: Pre Cal 12-2 lesson with notes 5th.notebookimages.pcmac.org/.../Uploads/Forms/Pre_Cal_Lesson_12-2.pdf · 2019. 9. 25. · Pre Cal 122 lesson with notes 5th.notebook 1 May 09, 2017

Pre Cal 12­2 lesson with notes 5th.notebook

2

May 09, 2017

May 9­6:50 AM

Dec 12­8:09 PM

Page 3: Pre Cal 12-2 lesson with notes 5th.notebookimages.pcmac.org/.../Uploads/Forms/Pre_Cal_Lesson_12-2.pdf · 2019. 9. 25. · Pre Cal 122 lesson with notes 5th.notebook 1 May 09, 2017

Pre Cal 12­2 lesson with notes 5th.notebook

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May 09, 2017

May 9­6:59 AM

May 9­6:56 AM

Example 1: Use the properties of limits to evaluate each limit.

1.  2. 3.

Page 4: Pre Cal 12-2 lesson with notes 5th.notebookimages.pcmac.org/.../Uploads/Forms/Pre_Cal_Lesson_12-2.pdf · 2019. 9. 25. · Pre Cal 122 lesson with notes 5th.notebook 1 May 09, 2017

Pre Cal 12­2 lesson with notes 5th.notebook

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May 09, 2017

Dec 12­8:30 PM

Use the properties of limits to evaluate each limit.

4. 5.  6.

Dec 12­8:39 PM

Page 5: Pre Cal 12-2 lesson with notes 5th.notebookimages.pcmac.org/.../Uploads/Forms/Pre_Cal_Lesson_12-2.pdf · 2019. 9. 25. · Pre Cal 122 lesson with notes 5th.notebook 1 May 09, 2017

Pre Cal 12­2 lesson with notes 5th.notebook

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May 09, 2017

May 9­7:07 AM

May 9­7:05 AM

Basically, direct substitution can be used to evaluate the limit of a function as long as the denominator, when evaluated, is not 0, or when there is a negative under the square root.

Example 2: Use direct substitution, if possible, to evaluate each limit. If not possible, explain why not.

7.  8. 9.

Page 6: Pre Cal 12-2 lesson with notes 5th.notebookimages.pcmac.org/.../Uploads/Forms/Pre_Cal_Lesson_12-2.pdf · 2019. 9. 25. · Pre Cal 122 lesson with notes 5th.notebook 1 May 09, 2017

Pre Cal 12­2 lesson with notes 5th.notebook

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May 09, 2017

Dec 12­8:51 PM

Use direct substitution, if possible, to evaluate each limit. If not possible, explain why not.

10. 11. 12.

May 9­7:10 AM

Page 7: Pre Cal 12-2 lesson with notes 5th.notebookimages.pcmac.org/.../Uploads/Forms/Pre_Cal_Lesson_12-2.pdf · 2019. 9. 25. · Pre Cal 122 lesson with notes 5th.notebook 1 May 09, 2017

Pre Cal 12­2 lesson with notes 5th.notebook

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May 09, 2017

Dec 13­5:33 AM

Indeterminate form: 0/0, when this result is obtained by direct substitution, you cannot determine the limit by this method. This just means you may or may not have a limit so try another method.

Factoring and dividing out any common factor:

Example 3: Evaluate each limit.

13. 14.

15. 16. 

May 9­7:11 AM

Page 8: Pre Cal 12-2 lesson with notes 5th.notebookimages.pcmac.org/.../Uploads/Forms/Pre_Cal_Lesson_12-2.pdf · 2019. 9. 25. · Pre Cal 122 lesson with notes 5th.notebook 1 May 09, 2017

Pre Cal 12­2 lesson with notes 5th.notebook

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May 09, 2017

Dec 13­5:46 AM

When dividing out a common factor, the result is a new function. The two functions yield the same function values for every x except when x = c. If two functions differ only at the value of c in their domain, then their limits as x approaches c are the same. This is because the value of a limit at a point is not dependent on the value of the function at that point. 

Another technique is to rationalize the numerator or denominator.

Example 4: Evaluate each limit.

17. 18. 19.

Dec 13­5:56 AM

In chapter 2, we learned at all even­degree power function have the same end behavior, and all odd­degree power functions have the same end behavior summarized by the chart.

Also in chapter 2 we learned that the end behavior of a polynomial function is determined by the end behavior of the power function related to its highest­powered term.

Note: notating that the limit of ∞ or ­∞ does not indicate that the limit exists, but instead describes the behavior of the function as increasing or decreasing without bound.

Also note: