Chapter 2 2012 Pearson Education, Inc. 2.3 Section 2.3 Continuity Limits and Continuity.
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Transcript of Chapter 2 2012 Pearson Education, Inc. 2.3 Section 2.3 Continuity Limits and Continuity.
2012 Pearson Education, Inc.
Chapter 2
2.3
Section 2.3
Continuity
Limits and Continuity
Slide 2.3- 2 2012 Pearson Education, Inc.
Quick Review
2
31
1 1
1
2
2 2
2
3 2 11. Find lim
4
2. Let int . Find each limit.
a lim b lim
c lim d 1
4 5, 23. Let
4 , 2
Find each limit.
a lim b lim
c lim d 2
x
x x
x
x x
x
x x
x
f x x
f x f x
f x f
x x xf x
x x
f x f x
f x f
Slide 2.3- 3 2012 Pearson Education, Inc.
Quick Review
2 2
In Exercises 4 – 6, find the remaining functions in the list of functions:
, , , .
2 1 14. , 1
5
5. , sin , domain of [0, )
f g f g g f
xf x g x
x x
f x x g f x x g
Slide 2.3- 4 2012 Pearson Education, Inc.
Quick Review
2
3
16. 1, , 0
7. Use factoring to solve 2 9 5 0
8. Use graphing to solve 2 1 0
g x x g f x xx
x x
x x
Slide 2.3- 5 2012 Pearson Education, Inc.
Quick Review
2
5 , 3In Exercises 9 and 10, let
6 8, 3
9. Solve the equation 4
10. Find a value of for which the equation
has no solution.
x xf x
x x x
f x
c f x c
Slide 2.3- 6 2012 Pearson Education, Inc.
Quick Review Solutions
2
31
1 1
1
2
2 2
2
2
2 1
no limi
3 2 11. Find lim
4
2. Let int . Find each limit.
a lim b lim
c lim d 1
4 5, 23. Let
4 , 2
Find each limit.
a lim b lim
l
t
1 2
i
1
c m
x
x x
x
x x
x
x x
x
f x x
f x f x
f x f
x x xf x
x x
f x f x
dno limit 22 f x f
Slide 2.3- 7 2012 Pearson Education, Inc.
Quick Review Solutions
2 2
2
In Exercises 4 – 6, find the remaining functions in the list of functions:
, , , .
2 1 14. , 1
5
5. , sin , domain of [0, )
2 3 4, 0 , 5
6 1 2 1
sin , 0 sin , 0
f g f g g f
xf
x xf g x x
x g xx x
f x
g f x xx x
g x x x f g x x x
x g f x x g
Slide 2.3- 8 2012 Pearson Education, Inc.
Quick Review Solutions
2
2
3
16. 1, , 0
7. Use factoring to solve 2 9 5 0
8. Use graphing to solve 2 1 0
11, 0 , 1
1
1, 5
2
0.453
xf x x f g x
g x x g f x xx
x x
x x
xxx
x
x
Slide 2.3- 9 2012 Pearson Education, Inc.
Quick Review Solutions
2
5 , 3In Exercises 9 and 10, let
6 8, 3
9. Solve the equation 4
10. Find a value of for which the equation
has no solution
1
Any in [1,2. )
x xf x
x x x
f x
c f
x
c
x c
Slide 2.3- 10 2012 Pearson Education, Inc.
What you’ll learn about
Continuity at a Point Continuous Functions Algebraic Combinations Composites Intermediate Value Theorem for Continuous Functions
…and whyContinuous functions are used to describe how a body moves through space and how the speed of a chemical reaction changes with time.
Slide 2.3- 11 2012 Pearson Education, Inc.
Continuity at a Point
Any function whose graph can be sketched in
one continuous motion without lifting the pencil is an
example of a continuous function.
y f x
Slide 2.3- 12 2012 Pearson Education, Inc.
Example Continuity at a Point
Find the points at which the given function is continuous and the points at
which it is discontinuous.o
Points at which is continuousfAt 0 x
At 6 x
At 0 < < 6 but not 2 3 c cPoints at which is discontinuousfAt 2xAt 0, 2 3, 6 c c c
0
lim 0
x
f x f
6
lim 6
x
f x f
lim
x c
f x f c
2
lim does not existxf x
these points are not in the domain of f
Slide 2.3- 13 2012 Pearson Education, Inc.
Continuity at a Point
Interior Point: A function is continuous at an interior point of its
domain if lim
Endpoint: A function is continuous at a left
endpoint or is continuous
x c
y f x c
f x f c
y f x
a
at a right endpoint of its domain if
lim or lim respectively.x a x b
b
f x f a f x f b
Slide 2.3- 14 2012 Pearson Education, Inc.
Continuity at a Point
If a function is , we say that
is and is a point of discontinuity of .
Note that need not be in the domain of .
f f
c f
c f
not continuous at a point
discontinuous at
c
c
Slide 2.3- 15 2012 Pearson Education, Inc.
Continuity at a Point
The typical discontinuity types are:
a) Removable (2.21b and 2.21c)
b) Jump (2.21d)
c) Infinite(2.21e)
d) Oscillating (2.21f)
Slide 2.3- 16 2012 Pearson Education, Inc.
Continuity at a Point
Slide 2.3- 17 2012 Pearson Education, Inc.
Example Continuity at a Point
There is an infinite discontinuity at 1.x
2
3Find and identify the points of discontinuity of
1y
x
[5,5] by [5,10]
Slide 2.3- 18 2012 Pearson Education, Inc.
Continuous Functions
A function is if and only if
it is continuous at every point of the interval. A
is one that is continuous at every
point of its domain. A continuous funct
continuous on an interval
continuous function
ion need not be
continuous on every interval.
Slide 2.3- 19 2012 Pearson Education, Inc.
Continuous Functions
2
2
2y
x
[5,5] by [5,10]
The given function is a continuous function because it is
continuous at every point of its domain. It does have a
point of discontinuity at 2 because it is not defined there.x
Slide 2.3- 20 2012 Pearson Education, Inc.
If the functions and are continuous at , then the
following combinations are continuous at .
1. Su ms:
2. Differences:
3. Products:
4. Constant multiples: , for any number
5. Quotients: , pr
f g x c
x c
f g
f g
f g
k f k
f
g
ovided 0g c
Properties of Continuous Functions
Slide 2.3- 21 2012 Pearson Education, Inc.
Composite of Continuous Functions
If is continuous at and is continuous at , then the
composite is continuous at .
f c g f c
g f c
Slide 2.3- 22 2012 Pearson Education, Inc.
Intermediate Value Theorem for Continuous Functions
0 0
A function that is continuous on a closed interval [ , ]
takes on every value between and . In other words,
if is between and , then for some in [ , ].
y f x a b
f a f b
y f a f b y f c c a b
Slide 2.3- 23 2012 Pearson Education, Inc.
The Intermediate Value Theorem for Continuous Functions is the reason why the graph of a function continuous on an interval cannot have any breaks. The graph will be connected, a single, unbroken curve. It will not have jumps or separate branches.
Intermediate Value Theorem for Continuous Functions