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Warm up Evaluate the limit x x 2 1 . lim x 1 x 3x 4 2 x x6 2 . lim h0 2 3 . lim x 1 x 2 2 4. lim x0 1 h 1 h x 25 x 5 Section 2.4 Continuity • SWBAT – Define continuity and its types Conceptual continuity 2.4 Continuity • This implies : 1. f(a) is defined 2. f(x) has a limit as x approaches a 3. This limit is actually equal to f(a) . Definition (cont’d) Types of discontinuity Removable Discontinuity: “A hole in the graph” (You can algebraically REMOVE the discontinuity) Types of discontinuity (cont’d) Infinite discontinuity: •Where the graph approaches an asymptote •It can not be algebraically removed jump discontinuity the function “jumps” from one value to another. Example • Where are each of the following functions discontinuous, and describe the type of discontinuity x3 1. f x 2 x x 12 x 9 x 20 2. f x x4 2 One-Sided Continuity • Continuity can occur from just one side: Continuity on an Interval • So far continuity has been defined to occur (or not) one point at a time. • We can also consider continuity over an entire interval at a time: • Continuous on an Interval: it is continuous at every point on that interval. Polynomials and Rational Functions • Write the interval where this function is 3 2 continuous. x 2x 1 lim : x2 5 3x 5 5 ( , ) ( , ) 3 3 Types of Continuous Function • We can prove the following theorem: • This means that most of the functions encountered in calculus are continuous wherever defined. 1. Lim f(x) x23. Lim f(x) x- 5. Lim f(x) x-2+ 7. f(2) 2. Lim f(x) x2+ 4. Lim f(x) x-26. Lim f(x) x0 8. f(-2) Assignment 8 • p. 126 1-31 odd • Quiz tomorrow – 2.1 through 2.4 Continuity