Lecture 4

1. Lecture 4 Continuity and Limits

2. Common Limit Calculation Can’t Simply Plug in x = 3 Since would be dividing by zero = = = = Effect of the calculations is to replace the function by one that is equal to it except at x = 3 and for which the limit can be calculated by “plugging in” the “3”

3. Geometry of the Calculation Graph of near x = 3

4. Limits and Piecewise Defined Functions Here f(1) exists and is equal to 4 but

5. limits from the left or right Does not exist

6. Left and Right Limits May Both Exist But Not Be Equal

7. Left and Right Limits of Piecewise Defined Functions = -1 = 7

8. A limit exists exactly when the limits from left and right bothexist and are equal. The limit is equal to this common value If exists and is equal to L then both and exist and both equal L Conversely, if both the left and right limits exist and they are equal to some number L then exists and is equal to L

9. Continuity When You Can Calculate Limits Simply By “Plugging In” The best possible situation is when we can do the following Sometimes it works, sometimes it doesn’t. Depends on f(x) and “a”

10. Won’t Work if Graph is BrokenSince Limit Does Not Exist. Here f(1) exists but does not

11. Wont Work if there is a “hole” in the graph at x = a since then either f(a) does not exist or it exists and is not equal to the limit

12. If f is a function and a is in its domain then these are equivalent • The graph of f(x) is “connected” at x = a in the sense that one could traverse it from the left of a to the right of a without encountering a “hole” or a “jump” at a. When this occurs we say that the function f is continuous at x = a

13. Points of Continuity • A function f is continuous if it is continuous at every point in its domain. • The function f can be continuous at some points and not at others • If f is continuous at every point in an interval (a, b) then we say that f is continuous on (a,b) • Continuous “from the right” and “from the left” are defined using right and left limits in the obvious way. • The function f is discontinuous at x = a if it is not continuous at x =a • Most functions one encounters in applications are discontinuous at only a “few” points.

14. What are some continuous functions? • If c is a constant then f(x) = c is continuous • f(x) = x is continuous • If f(x) and g(x) are continuous on an interval (a,b) then f/g is continuous at all x in (a,b) for which g(x) is not zero • If f(x) and g(x) are continuous and f(g(x)) is defined on an interval (a,b) the f(g) is continuous on (a,b) • [For future reference] If f is differentiable on an interval (a, b) then f is continuous on (a,b)

15. Philosophically: • This says that functions with rules described by single formulas are continuous • For functions defined piecewise with a single formula on each interval it says that the functions are continuous on the each of the intervals – end points have to be checked (anything can happen)

16. Calculating Rule For Limits With Continuous Functions If f is a continuous function, g is a function and g(x) is in the domain of f for x “near” a then: Ifg is itself continuous at a and g(a) is in the domain of f then

17. Fractional Exponents For n even is defined and continuous for x > 0 For n odd is defined and continuous for all x

18. Example = = > 0 f(g(1) =