Lecture 4 Continuity and Limits

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”

Geometry of the Calculation Graph of near x = 3

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

limits from the left or right Does not exist

Left and Right Limits May Both Exist But Not Be Equal

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

A limit exists exactly when the limits from left and right both exist 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

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”

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

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

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

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.

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)

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)

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

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

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