# 1-3 Continuity, End Behavior, and Limits

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1-3 Continuity, End Behavior, and Limits Name ___________________________

We have already found the domain and range using the graph of a function. Now, we will:

1. use limits to determine the continuity of a function,

2. apply the Intermediate Value Theorem to continuous functions, and

3. use limits to describe end behavior of functions.

1. Continuity

• The graph of a continuous function has no ________, __________, or _________________. You can trace the graph of a continuous function _________________________________________________.

• One condition for a function f(x) to be continuous at a point where x=c is that the function must approach a unique function value (the value of f(x) or the y-coordinate) as x-values approach c from the left and right sides.

• The concept of approaching a value without necessarily ever reaching it is called a ___________.

[pic]

• To understand what it means for a function to be continuous from an algebraic perspective, it helps to examine graphs of functions that are not continuous – discontinuous functions.

[pic]

Notice what happens at these discontinuities as the point at which they’re discontinuous is approached from the left and right…

• To show that a function is continuous from its function, we must show all of the following:

[pic]

• Example: Determine whether f(x) = 2x2 – 3x – 1 is continuous at x = 2. Justify using the test above.

1. Does f(2) exist?

2. Does [pic] exist?

|x | | | |2.0 | | | |

|f(x) | | | | | | | |

Not all graphs have limits at infinity.

• Example: Use the graph below to describe its end behavior. Support the conjecture numerically.

Analyze Graphically Support Numerically

x |-10,000 |-1,000 |-100 |0 |100 |1,000 |10,000 | |f(x) | | | | | | | | |[pic]

• Example: Use the graph below to describe its end behavior. Support the conjecture numerically.

Analyze Graphically Support Numerically

x |-10,000 |-1,000 |-100 |0 |100 |1,000 |10,000 | |f(x) | | | | | | | | |

CW/HW Assignment: ________________________

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