1 4 Continuity and One Sided Limits Objective
1. 4 Continuity and One. Sided Limits Objective: Determine continuity at a point and on an open interval; determine one-sided limits and continuity on a closed interval. Miss Battaglia AB/BC Calculus
Continuity � What does it mean to be continuous? Below are three values of x at which the graph of f is NOT continuous At all other points in the interval (a, b), the graph of f is uninterrupted and continuous f(c) is not defined does not exist
Definition of Continuity at a Point: A function f is continuous at c if the following three conditions are met. 1. f(c) is defined 2. exists 3. Continuity on an Open Interval: A function is continuous on an open interval (a, b) if it is continuous at each point in the interval. A function that is continuous on the entire real line (-∞, ∞) is everywhere continuous.
Discontinuities � Removable (f can be made continuous by appropriately defining f(c)) & nonremovable. Nonremovable Discontinuity Removable Discontinuity
Continuity of a Function � Discuss the continuity of each function
One-Sided Limits and Continuity on a Closed Interval • Limit from the right • Limit from the left • One-sided limits are useful in taking limits of functions involving radicals (Ex: if n is an even integer)
A One-Sided Limit � Find the limit of -2 from the right. as x approaches
The Greatest Integer Function �
Theorem 1. 10: The Existence of a Limit Let f be a function and let c and L be real numbers. The limit of f(x) as x approaches c is L iff and Definition of Continuity on a Closed Interval A function f is continuous on the closed interval [a, b] if it is continuous on the open interval (a, b) and The function f is continuous from the right at a and continuous from the left at b.
Continuity on a Closed Interval � Discuss the continuity of
By Thm 1. 11, it follows that each of the following functions is continuous at every point in its domain.
Intermediate Value Theorem � Consider a person’s height. Suppose a girl is 5 ft tall on her thirteenth bday and 5 ft 7 in tall on her fourteenth bday. For any height, h, between 5 ft and 5 ft 7 in, there must have been a time, t, when her height was exactly h. � The IVT guarantees the existence of at least one number c in the closed interval [a, b]
An Application of the IVT � Use the IVT to show that the polynomial function f(x)=x 3 + 2 x – 1 has a zero in the interval [0, 1]
Classwork/Homework �AB: Page 78 #27 -30, 35 -51 odd, 69 -75 odd, 78, 79, 83, 91, 99 -102 �BC: Page 78 #3 -25 every other odd, 31, 33, 34, 35 -51 every other odd, 61, 63, 69, 78, 91, 99 -103
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