Functions, Limits, and Continuity

1. Functions, Limits, and Continuity Section 1.3

2. Functions, Limits, and Continuity • Limit • Fundamental concept of calculus • Separates calculus from algebra • Two types • Toward infinity • Toward a point

3. Functions, Limits, and Continuity • Limit toward infinity • As the value of the input variable grows large (or small) without bound, what happens to the value of the output variable • “End behavior of the function”

4. Functions, Limits, and Continuity Y= Y1 = 6X+2 GRAPH

5. Functions, Limits, and Continuity “Limit as z goes to infinity of g(z) is infinity” “Limit as z goes to negative infinity of g(z) is negative infinity”

6. Functions, Limits, and Continuity

7. Functions, Limits, and Continuity Y= Y1 = 9/(1+e^(-x)) GRAPH

8. Functions, Limits, and Continuity

9. Functions, Limits, and Continuity • Numerically evaluating limits toward infinity • Calculate value of output variable for progressively larger (smaller) values of the input variable TBLSET Indpnt: Ask Depend: Auto TABLE

10. Functions, Limits, and Continuity 8.9996 8.9777 8.5732 6.5795 “Limit as x goes to infinity of r(x) is 9” 1 3 5 10

11. Functions, Limits, and Continuity 8.9996 8.9777 8.5732 6.5795 1 3 5 10

12. Functions, Limits, and Continuity

13. Functions, Limits, and Continuity

14. Functions, Limits, and Continuity

15. Functions, Limits, and Continuity

16. Functions, Limits, and Continuity

17. Functions, Limits, and Continuity • Toward a point • As the input variable gets arbitrarily close to a certain value, what happens to the value of the output variable? • Limit must be the same from both sides

18. Functions, Limits, and Continuity “Limit as z goes to 4 from the right of g(z) is 26” 26 4 “Limit as z goes to 4 from the left of g(z) is 26”

19. Functions, Limits, and Continuity 26

20. Functions, Limits, and Continuity

21. Functions, Limits, and Continuity • Continuity • A function is continuous if it is defined at every input with no breaks or sudden jumps in the output value • “Function can be drawn without lifting your pen”

22. Functions, Limits, and Continuity Yes

23. Functions, Limits, and Continuity No

24. Functions, Limits, and Continuity Yes

25. Functions, Limits, and Continuity 1993 Does the limit exist at 1993? Is the function continuous? Is the function continuous from 1993 on?

26. Functions, Limits, and Continuity • Application: Continuously compounded interest P = principal r = interest rate of a particular term t = number of terms principal is invested A = accumulated value

27. Functions, Limits, and Continuity • Application: Continuously compounded interest P = principal r = annual interest rate t = number of years principal is invested A = accumulated value

28. Functions, Limits, and Continuity • Term compounding • Compounding matches the term of the rate • e.g., annual interest rate with annual compounding

29. Functions, Limits, and Continuity • General compounding • Compounding happens n times per term

30. Functions, Limits, and Continuity • Example: Semi-annual compounding • Happens 2 times per year

31. Functions, Limits, and Continuity • Continuous compounding • Infinite number of periods per term

32. Functions, Limits, and Continuity t = 15 years P = $1,000 r = 8% per year Annual Monthly Continuous

33. Functions, Limits, and Continuity • In-Class