Objectives

1. Objectives • Informal definition of limits • Limit laws • Skip formal definition of limit (epsilon, delta) • Basic computational tools

2. Future Applications • Limits in 3 basic calculus problems: • Tangent line problem: Find tangent line to curve at given point • Area problem: Find area under curve • Instantaneous velocity problem: If s=position=f(t), t=time, find instantaneous velocity at given time

3. Rough definition • Write if f(x) can be made sufficiently close to L by making x sufficiently close (but not equal to) a • Think of always needing (x,f(x)) to be in narrow ribbon around y=L for x close to a (but not equal to a)

4. Remarks • Important: For limit at x=a, value of f at a not relevant • E.g., If f(x) = 25 for x0 and f(0) = 347, then limit of f as x approaches 0 is 25 • Find

5. Harder Examples • Compute the following limits:

6. One-sided limits • Consider f(x)=x/|x|. Then f(x) tends to 1 as x tends to 0 from the right , and it tends to -1 as x tends to 0 from the left • This function has a jump at 0 so (two-sided) limit doesn’t exist even though both one-sided ones do

7. Definition of one-sided limits • Definition of right, left limits similar to two-sided limit except only use values of f to one side of a • Remark: for (two-sided) limit to exist the two one-sided limits must exist and agree

8. Left hand limit

9. Limit does not exist: oscillation

10. The limiting process is demonstrated by first displaying blue dots approaching c in the domain. These are connected to the functional values via dotted lines through the function's graph. The corresponding functional values are displayed as red dots on the vertical axis. As the blue dots converge on 0, the red dots converge on the limit. Convergent limit from both sides

11. Infinite Limits • Infinite-valued limits (f blowing up):

12. An infinite limit

13. Examples Line x=a a vertical asymptote if each one-sided limit approachs 

14. Limit laws • Suppose that a is a constant and that the limit of f and g at c exists. Then

15. Polynomials and rational functions • limxcx = c. • limxcx2 = limxcx limxcx = c2 • More generally:

16. Examples

17. Root law

18. Comparison Theorem

19. Damped Oscillation

20. Sandwich Theorem

21. Related Examples • Compute the following limits

22. Limits at infinity

23. Damped Oscillation at infinity

24. More Examples

25. Limits of rational functions at infinity

26. Examples

27. Logistic Growth • Suppose N(t) population at time t. Under logistic growth model:

28. Logistic Graph

29. Example

30. Exponential Population Growth • Unlimited population growth, similar to compound interest on bank account (or credit card account!) • Exponential population growth is only appropriate for describing the initial colonization of empty habitats.

31. Exponential Growth Graph

32. Exponential tidbit • If human population of earth were to increase at this rate indefinitely: • By A.D. 3530 the total mass of human flesh and blood would equal the mass of the earth • By A.D. 6826 the total mass of human flesh and blood would equal the mass of the known universe • Source: “All I really needed to know I learned in Kindergarten”, Robert Folghum, 1986