Warm up
Evaluate the limit
2
211. lim
3 4
x
x x
x x 0
1 12. lim
h
h
h
2
21
63. lim
2x
x x
x
0
4. lim25 5 x
x
x
Section 2.4
Continuity• 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.
12x
f xx x
2 9 202.
4x x
f 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
