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Splash Screen
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Lesson Menu
Five-Minute Check Then/Now New Vocabulary Key
Concept Limits Key
Concept Types of Discontinuity Concept Summary
Continuity Test Example 1 Identify a Point of
Continuity Example 2 Identify a Point of
Discontinuity Key Concept Intermediate Value
Theorem Example 3 Approximate Zeros Example
4 Graphs that Approach Infinity Example
5 Graphs that Approach a Specific Value Example
6 Real-World Example Apply End Behavior
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5Minute Check 1
Use the graph of f (x) to find the domain and
range of the function.
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5Minute Check 2
Use the graph of f (x) to find the y-intercept
and zeros. Then find these values algebraically.
A. y-intercept 9, zeros 2 and
3 B. y-intercept 8, zeros 1.5 and
3 C. y-intercept 9, zeros 1.5 and
3 D. y-intercept 8, zero 1
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5Minute Check 3
Use the graph of y x 2 to test for symmetry
with respect to the x-axis, y-axis, and the
origin.
A. y-axis B. x-axis C. origin D. x- and y-axis
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Then/Now
You found domain and range using the graph of a
function. (Lesson 1-2)

• Use limits to determine the continuity of a
  function, and apply the Intermediate Value
  Theorem to continuous functions.
• Use limits to describe end behavior of functions.

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Vocabulary

• continuous function
• limit
• discontinuous function
• infinite discontinuity
• jump discontinuity
• removable discontinuity
• nonremovable discontinuity
• end behavior

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Key Concept 1
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Key Concept 2
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Concept Summary 1
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Example 1
Identify a Point of Continuity
Check the three conditions in the continuity test.
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Example 1
Identify a Point of Continuity
Construct a table that shows values of f(x)
approaching from the left and from the right.
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Example 1
Identify a Point of Continuity
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Example 1
Identify a Point of Continuity
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Example 1
Determine whether the function f (x) x 2 2x
3 is continuous at x 1. Justify using the
continuity test.
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Example 2
Identify a Point of Discontinuity
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Example 2
Identify a Point of Discontinuity
2. Investigate function values close to f(1).
The pattern of outputs suggests that for values
of x approaching 1 from the left, f (x) becomes
increasingly more negative. For values of x
approaching 1 from the right, f (x) becomes
increasing more positive.
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Example 2
Identify a Point of Discontinuity
3. Because f (x) decreases without bound as x
approaches 1 from the left and f (x) increases
without bound as x approaches 1 from the right,
f (x) has infinite discontinuity at x 1. The
graph of f (x) supports this conclusion.
Answer f (x) has an infinite discontinuity at x
1.
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Example 2
Identify a Point of Discontinuity
Therefore f (x) is discontinuous at x 2.
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Example 2
Identify a Point of Discontinuity
2. Investigate function values close to f (2).
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Example 2
Identify a Point of Discontinuity
Answer f (x) is not continuous at x 2, with a
removable discontinuity.
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Example 2
A. f (x) is continuous at x 1. B. infinite
discontinuity C. jump discontinuity D. removable
discontinuity
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Key Concept 3
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Example 3
Approximate Zeros
Investigate function values on the interval -2,
2.
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Example 3
Approximate Zeros
Because f (-1) is positive and f (0) is negative,
by the Location Principle, f (x) has a zero
between -1 and 0. The value of f (x) also changes
sign for 1,2. This indicates the existence of
real zeros in each of these intervals. The graph
of f (x) supports this conclusion.
Answer There are two zeros on the interval, 1
lt x lt 0 and 1 lt x lt 2.
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Example 3
Approximate Zeros
B. Determine between which consecutive integers
the real zeros of f (x) x 3 2x 5 are
located on the interval 2, 2.
Investigate function values on the interval 2,
2.
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Example 3
Approximate Zeros
Because f (-2) is negative and f (1) is
positive, by the Location Principle, f (x) has a
zero between 2 and 1. This indicates the
existence of real zeros on this interval. The
graph of f (x) supports this conclusion.
Answer 2 lt x lt 1.
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Example 3
A. Determine between which consecutive integers
the real zeros of f (x) x 3 2x 2 x 1 are
located on the interval 4, 4.
A. 1 lt x lt 0 B. 3 lt x lt 2 and 1 lt x lt 0 C. 3
lt x lt 2 and 0 lt x lt 1 D. 3 lt x lt 2, 1 lt x lt
0, and 0 lt x lt 1
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Example 3
B. Determine between which consecutive integers
the real zeros of f (x) 3x 3 2x 2 3 are
located on the interval 2, 2.
A. 2 lt x lt 1 B. 1 lt x lt 0 C. 0 lt x lt 1 D. 1 lt
x lt 2
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Example 4
Graphs that Approach Infinity
Use the graph of f(x) x 3 x 2 4x 4 to
describe its end behavior. Support the conjecture
numerically.
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Example 4
Graphs that Approach Infinity
Analyze Graphically
Support Numerically
Construct a table of values to investigate
function values as x increases. That is,
investigate the value of f (x) as the value of x
becomes greater and greater or more and more
negative.
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Example 4
Graphs that Approach Infinity
The pattern of output suggests that as x
approaches 8, f (x) approaches 8 and as x
approaches 8, f (x) approaches 8.
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Example 4
Use the graph of f (x) x 3 x 2 2x 1 to
describe its end behavior. Support the conjecture
numerically.
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Example 5
Graphs that Approach a Specific Value
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Example 5
Graphs that Approach a Specific Value
Analyze Graphically
Support Numerically
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Example 5
Graphs that Approach a Specific Value
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Example 5
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Example 6
Apply End Behavior
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Example 6
Apply End Behavior
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Example 6
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End of the Lesson