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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

Homework Quiz –Write the following work and answer of the following questions 3, 9, 17, 27

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.

Vocabulary 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!!!!

Key Concept 1

Key Concept 2

Example 1 Identify a Point of Continuity Check the three conditions in the continuity test. Determine whether is continuous at. Justify using the continuity test. 1.Does exist? 2.Does exist? 3.Does ?

Use a graphics calculator to find the point quickly

Example 1 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.

Example 2 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.

Example 2 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.

Example 2 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.

Example 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.

Example 2 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. Answer: f (x) is discontinuous at x = 2 with a removable discontinuity. 4

Example 2 A.f (x) is continuous at x = 1. B.infinite discontinuity C.jump discontinuity D.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.

Example 3 Approximate Zeros Investigate function values on the interval [  2, 2]. A. Determine between which consecutive integers the real zeros of are located on the interval [–2, 2].

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.

Example 4 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.

Example 5 Graphs that Approach a Specific Value Use the graph of to describe its end behavior. Support the conjecture numerically.

Example 5 Use the graph of to describe its end behavior. Support the conjecture numerically. A. B. C. D.

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