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Chapter 2

Chapter 2. Limits and Continuity. Introduction. Economic Injury Level (EIL)- text p. 58. 2.1: Rates of Change and Limits. Average Speed The average speed during an interval of time is Example : If a rock falls from a cliff, what is its average speed after the first 2 seconds?

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Chapter 2

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  1. Chapter 2 Limits and Continuity

  2. Introduction • Economic Injury Level (EIL)- text p. 58

  3. 2.1: Rates of Change and Limits Average Speed The average speed during an interval of time is • Example: If a rock falls from a cliff, what is its average speed after the first 2 seconds? (Distance for a falling object equation: d = 16t2) = = = = 32 ft/s

  4. Instantaneous Speed (speed at an exact time) • Example: If a rock falls from a cliff, what is its instantaneous speed at 2 seconds? To answer this question, we will find the AVERAGE speed of the rock between 2 seconds and slightly more than 2 seconds (we’ll call it “2+h” seconds, where his approximately zero). (Recall: Distance for a falling object equation: d = 16t2) = = = = … =64 At 2 seconds, the rock is travelling at 64 ft/s.

  5. Instantaneous change using the table in the graphing calculator (Calculator may only be used to check.) • Enter the following in Y1: • TableSet: start at x =1 Δx= .1 (look at table) • Change to: start at x =.1 Δx= .01 (look at table) • Change to: start at x =.01 Δx= .001 (look at table) • What value is y getting close to? As h approaches 0, y has the limiting value of 64.

  6. Definition of LIMIT: L a

  7. The Definition of Limit L a

  8. Basic Limits (Substitution) • limx4 2x – 5 = • limx-3 x2 = • limxcos x = • limx1sin =

  9. Indeterminate Forms Indeterminate forms occur when substitution in the limit results in 0/0. In such cases either factor or rationalize the expressions. Notice form Ex. Factor and cancel common factors

  10. More Examples

  11. More Examples Evaluate the following:. • Existence of a limit does not depend on whether or not the function is defined at c.

  12. Example: Find using the graphing calculator.

  13. (Found on text p. 61) Limit Theorems

  14. More Examples

  15. Find the limit:

  16. One-Sided Limits If limxc+f(x) = limxc-f(x) = L then, limxcf(x)=L (Again, L must be a fixed, finite number.) Basically, the y-value the function approached from the left side of a particular x-value must be the same y-value the function approaches from the right!

  17. One-Sided Limit Example 1. Given Find Find

  18. One-Sided Limits Example: f(4) = f(2) =

  19. One-Sided Limits f(0) = f(4) = f(3) = f(6) =

  20. But

  21. One-Sided Limits

  22. Limits from Tables • For each of the following tables, find the right-hand, left-hand, and "regular" limits at the center value of the table. If no limit exists, then state that.

  23. Limits from Tables

  24. Summary • The limit of a function at x = c does not depend on the value of f(c). • The limit only exists when the limit from the right equals the limit from the left and the value is a FIXED, FINITE #! • A common limit you need to memorize: • Limits fail to exist: (ask for pictures) 1. Unbounded behavior – not finite 2. Oscillating behavior – not fixed 3. - fails def of limit • Limits can be found algebraically (substitution), graphically, and from a table.

  25. HOMEWORK: p.

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