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Round and round in calc. we go!. Get out your assignment. Warm up. Evaluate the limit. Section 2.4 Continuity. SWBAT Define continuity and its types. Conceptual continuity. 2.4 Continuity. This implies : f ( a ) is defined f ( x ) has a limit as x approaches a - PowerPoint PPT Presentation
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Warm upEvaluate the limit
2
211. lim
3 4
x
x xx x 0
1 12. limh
hh
2
21
63. lim2x
x xx
0
4. lim25 5 x
xx
Section 2.4Continuity
• SWBAT– Define continuity and its types
Conceptual continuity
2.4 Continuity
• This implies :1. f(a) is defined2. f(x) has a limit as x approaches a3. 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
2
31.12
xf xx x
2 9 202.4
x xf xx
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 continuous.3 2
2
2 1lim :5 3x
x xx
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)x2-
2. Lim f(x)x2+
3. Lim f(x)x-
4. Lim f(x)x-2-
5. Lim f(x) x-2+
6. Lim f(x)x0
7. f(2) 8. f(-2)
Assignment 8
• p. 126 1-31 odd
• Quiz tomorrow – 2.1 through 2.4 Continuity