BC 1-2 Limits 4 Name: Limits and Continuity The concept of limit allows us to replace our informal definition of continuity with one that is mathematically precise. Definition: A function is continuous at a point x = a in the domain of f if and only if lim x a . xa In the case f is continuous at every point in some set S, we say f is continuous on S. If f is not continuous at x = a, we say the function is discontinuous at x = a. (1) (a) Determine any point(s) of discontinuity (x-values) for each of the following functions. x (b) x x 1 x 2 (c) 2 (2) (3) BC 1-2 1 g x x x 2 3x 3 x2 9 , x3 Suppose x x 3 . 4, x 3 (a) Show that is discontinuous at x = 3. Justify by using the definition for continuity listed above. (b) How should (3) be defined instead to make continuous at that point? (a) Give an example of a function with a removable discontinuity (i.e., a hole) at x = 2. (b) Lim 4.1 Give an example of a function with a jump discontinuity (i.e., a jump or a break forming two separate sections) at x = 2. (4) (5) (6) (7) BC 1-2 3ax 7, x 1 Let h x 2 . Find the value(s) of a so that h will be continuous at x a, x 1 x 1 . (Use the definition and limits!) Define the value of g ( 2) so that g will be continuous if g x x 2 3x 10 x2 2 x . 3x 2, x 2 x 2 , use the definition of continuity and limits to determine whether If k x 5, 2 x , x2 or not k is continuous at x = 2. Find each limit. Approximate and guess if necessary. ex sin x (a) lim (b) lim x x x x Lim 4.2 (c) lim x ln x x Intermediate Value Property: A function f satisfies the Intermediate Value Property (IVP) on the interval [a,b] if and only if for each number k between f (a ) and f (b) , there exists a value c [ a, b] such that f (c ) k . (8) Which of the following functions satisfy the IVP on the given interval? a. The function below on [2,3] ? 2 1 b. The function f ( x ) c. Define f on (0,1) by On [2, 0] ? On [0, 3] ? 3 1 on [1,1] ? x On [2,10] ? p 1 if x is rational and x in lowest terms q f ( x) q 0 if x is irrational . 1 3 Does f satisfy the IVP on , . 2 2 The Intermediate Value Theorem: If a function f is continuous on the closed interval [a, b] , then f satisfies the Intermediate Value Property (IVP) on the interval [a,b]. Alternately, If a function f is continuous on the closed interval [a, b] , then for each number k between f (a ) and f (b) , there exists a value c [ a, b] such that f (c ) k . BC 1-2 Lim 4.3