Download 1-3 Continuity_End_Behavior_and_Limits

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
• Homework Quiz
– Write the following work and answer of the
following questions
• 3, 9, 17, 27
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.
• continuous function
• limit
• discontinuous function
• infinite discontinuity
• jump discontinuity
• removable discontinuity
• nonremovable discontinuity
• end behavior
Mr. Schrauben show
students open forum
where the book is!!!!
Identify a Point of Continuity
Determine whether
is continuous at
. Justify using the continuity test.
Check the three conditions in the continuity test.
1. Does
exist?
2. Does
exist?
3. Does
?
• Use a graphics calculator to find the point
quickly
Determine whether the function f(x) = x 2 + 2x – 3 is
continuous at x = 1. Justify using the continuity
test.
A. continuous;
f(1)
B. Discontinuous; the function is undefined at x = 1
because
does not exist.
Identify a Point of Discontinuity
A. Determine whether the function
is
continuous at x = 1. Justify using the continuity
test. If discontinuous, identify the type of
discontinuity as infinite, jump, or removable.
Identify a Point of Discontinuity
2. Investigate function values close to f(1). You can
get this table on a graphics calculator
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.
Therefore,
does not exist.
Identify a Point of Discontinuity
B. Determine whether the function
is
continuous at x = 2. Justify using the continuity
test. If discontinuous, identify the type of
discontinuity as infinite, jump, or removable.
1. Because
is undefined, f(2) does not exist.
Therefore f(x) is discontinuous at x = 2.
Identify a Point of Discontinuity
2. Investigate function values close to f(2).
The pattern of outputs suggests that f(x)
approaches 0.25 as x approaches 2 from each
side, so
.
Identify a Point of Discontinuity
3. Because
exists, but f(2) is undefined,
f(x) has a removable discontinuity at x = 2. The
graph of f(x) supports this conclusion.
4
Answer: f(x) is discontinuous at x = 2 with a
removable discontinuity.
Determine whether the function
is continuous at x = 1. Justify using the continuity
test. If discontinuous, identify the type of
discontinuity as infinite, jump, or removable.
A. f(x) is continuous at x = 1.
B. infinite discontinuity
C. jump discontinuity
D. removable discontinuity
Approximate Zeros
A. Determine between which consecutive integers
the real zeros of
are located on the
interval [–2, 2].
Investigate function values on the interval [-2, 2].
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.
Use the graph of
f(x) = x 3 + x 2 – 2x + 1 to
describe its end behavior.
Support the conjecture
numerically.
A.
B.
C.
D.
Graphs that Approach a Specific Value
Use the graph of
to describe its end
behavior. Support the conjecture numerically.
Use the graph of
to describe its end
behavior. Support the conjecture numerically.
A.
B.
C.
D.