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More Limits

More Limits. Objective: SWBAT Evaluate one sided and two sided limits, and determine if a limit exists graphically. Do Now. For the graph of f pictured, evaluate the expressions below. Do Now. Agenda. Do Now Review Limits – How to write them and evaluate them Graphically Limits Numerically

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More Limits

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  1. More Limits Objective: SWBAT Evaluate one sided and two sided limits, and determine if a limit exists graphically

  2. Do Now • For the graph of f pictured, evaluate the expressions below

  3. Do Now

  4. Agenda • Do Now • Review Limits – How to write them and evaluate them Graphically • Limits Numerically • One Sided Limits • Non-existent limits Objective: SWBAT Evaluate one sided and two sided limits, and determine if a limit exists graphically

  5. Limits • “The limit of f(x) as x approaches (or goes to) c” • This means: What function value does the function approach as x gets closer and closer to c from both sides.

  6. Agenda • Do Now • Review Limits – How to write them and evaluate them Graphically • Limits Numerically • One Sided Limits and Non-existent limits Objective: SWBAT Evaluate one sided and two sided limits, and determine if a limit exists graphically

  7. Evaluating Limits Numerically • We can test to see what number a function is getting closer and closer to numerically. • Calculators: put the following function in for Y1 • Go to Table Set, and change the independent variable(x) to ‘ask’ and the dependent variable to ‘auto’

  8. Limits - Numerically • Let’s find out what the limit of this function is as we approach x=4. Set up this table:

  9. Similarly • Check the other side!

  10. Now you try • As a table, I’ll give you a function, and an x value to approach. • Sketch a graph of the function (including any holes or asymptotes) • Create a table with values approaching from both sides. • Evaluate the limit • Put all of it on your spot on the board after checking with me.

  11. Some limits don’t exist!!!

  12. Non-Existent Limits • If a function does not converge (get closer and closer to one value) at a particular x value, we say it’s limit does not exist • i.e.

  13. Remember:One Sided Limits • We can evaluate a limit from one direction. • This is called a one sided limit The limit of the function as x approaches c from the left The limit of the function as x approaches c from the right

  14. Example

  15. Numerically-the rules!!! • If both sides of the table get closer to that value then it is that value • If both sides of the table get larger and larger in the same direction then, the limit is positive or negative infinity • If both sides of the table get larger in different directions, then the limit does not exist.

  16. Eeeeek! What if I plug in and the universe explodes? If you plug in you get 1/0 and the universe explodes. In this case, you must solve numerically, or solve graphically. Good news! It can be done in your head. • For example, what happens to 1/x as you get closer and closer to 0, from the left. (x=-.1, -.01, -.001 etc). • What about from the right? (x=.1, .01, .001 etc) • This limit does not exist.

  17. You try, in your head

  18. Together adding limits Page 87 #2

  19. Homework • Anton problem set P. 76 #4 (quick check) P. 87 (3-5, 9,10, 12, 29-32)

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