objectives: 1.be able to define continuity by determining if a graph is continuous. 2.be able to...
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Objectives:1. Be able to define continuity by determining if a graph is
continuous.2. Be able to identify and find the different types of
discontinuities that functions may contain.3. Be able to determine if a function is continuous on a
closed interval.4. Be able to determine one-sided limits and continuity on
a closed interval.
Critical Vocabulary:Limit, Continuous, Continuity, Composite Function
I. Continuity
Continuous: To say that a function f is continuous at x = c there is no interruption in the graph of f at c
This means a graph will contain no _____, _____, or _____
Simple Terms: If you ever have to lift your pencil to sketch a graph, then it is not continuous.
I. Continuity
What Causes discontinuity?
1. The function is not defined at c.
This is an example of a ____ in the graph at ___
Concept: The function is not defined at c.
__________________
Let’s look at at f(x) = ½x - 2
I. Continuity
What Causes discontinuity?
2. The limit of f(x) does not exist at x = c
This is an example of a ____ in the graph at ______
Concept: The limit does not exist at x = c
Let’s look at at
37
313
1)(
xx
xxxf
I. Continuity
What Causes discontinuity?
3. The limit of f(x) exists at x = c but is not equal to f(c).
This is an example of a _____ in the graph
Concept: The behavior (limit) and where its defined (f(c)) are __________________
Let’s look at the first graph again
)()(lim cfxfcx
What is the limit as x approaches -2?
What is f(-2)?
A function f is continuous at c if the following three conditions are met:
1. ___________________________
2. ___________________________
3. ___________________________
I. Continuity
Continuous: To say that a function f is continuous at x = c there is no interruption in the graph of f at c
This means a graph will contain no HOLES, JUMPS, or GAPS
Simple Terms: If you ever have to lift your pencil to sketch a graph, then it is not continuous.
Objectives:1. Be able to define continuity by determine if a graph is
continuous.2. Be able to identify and find the different types of
discontinuities that functions may contain.3. Be able to determine if a function is continuous on a
closed interval.4. Be able to determine one-sided limits and continuity on
a closed interval.
Critical Vocabulary:Limit, Continuous, Continuity, Composite Function
II. Discontinuities
When you are asked to “discuss the continuity” of each function, you are really being asked to describe any place where the graph is discontinuous.
Discontinuity is broken into 2 Categories:
1. Removable:A discontinuity is removable if you COULD define f(c).
II. Discontinuities
When you are asked to “discuss the continuity” of each function, you are really being asked to describe any place where the graph is discontinuous.
Discontinuity is broken into 2 Categories:
1. Removable:A discontinuity is removable if you COULD define f(c).
2. Non-Removable:
A discontinuity is non-removable if you CANNOT define f(c).
II. Discontinuities
When you are asked to “discuss the continuity” of each function, you are really being asked to describe any place where the graph is discontinuous.
Discontinuity is broken into 2 Categories:
1. Removable:A discontinuity is removable if you COULD define f(c).
2. Non-Removable:
A discontinuity is non-removable if you CANNOT define f(c).
Example 1:1
1)(
2
x
xxf What is the Domain?
II. Discontinuities
When you are asked to “discuss the continuity” of each function, you are really being asked to describe any place where the graph is discontinuous.
Discontinuity is broken into 2 Categories:
A discontinuity is removable if you COULD define f(c).
A discontinuity is non-removable if you CANNOT define f(c).
Example 2: 9
3)(
2
x
xxf What is the Domain?
II. Discontinuities
When you are asked to “discuss the continuity” of each function, you are really being asked to describe any place where the graph is discontinuous.
Discontinuity is broken into 2 Categories:
1. Removable:A discontinuity is removable if you COULD define f(c).
2. Non-Removable:
A discontinuity is non-removable if you CANNOT define f(c).
Example 3: Discuss the continuity of the composite function f(g(x))
xxf
1)( 1)( xxg
II. Discontinuities
Example 5: Graph the piecewise function, then determine on which intervals the graph is continuous.
02
01)( 2 xx
xxxf
III. Closed Intervals
Example 5: Discuss the continuity on the closed interval.
2,14
1)(
2
x
xf
Closed Interval: Focusing on specific portion (domian) of a graph. [a, b]
1. Page 237 #23-43 odd, 49-55 odd, 61, 63, 77
2. Page 236 #1-17 odd, 79, 88
Objectives:1. Be able to define continuity by determine if a graph is
continuous.2. Be able to identify and find the different types of
discontinuities that functions may contain.3. Be able to determine if a function is continuous on a
closed interval.4. Be able to determine one-sided limits and continuity on
a closed interval.
Critical Vocabulary:Limit, Continuous, Continuity, Composite Function
IV. One-Sided Limits
What does a One-Sided look like?
Lxfcx
)(lim c
Lxfcx
)(lim c Approach from the right only
Lxfcx
)(lim c Approach from the left only
IV. One-Sided Limits
Example 1: Graph then find the limits
24)( xxf
What’s the domain?
x
f(x)
____)(lim2
xfx
____)(lim2
xfx
____)(lim0
xfx
____)(lim2
xfx
____)(lim2
xfx
____)(lim2
xfx
____)(lim2
xfx
IV. One-Sided Limits
Example 1: Graph then find the limits
14
14)( 2 xxx
xxxf
x
f(x)
____)(lim1
xfx
____)(lim1
xfx
____)(lim1
xfx
Is this graph continuous?
1. Page 237 #23-43 odd, 49-55 odd, 61, 63, 77
2. Page 236 #1-17 odd, 79, 88