Bell Ringer
Write a paragraph answering the following
questions.
What is a zero of a function? How do we find it?
1.
Continuity, End Behavior, and
Limits
LEARNING TARGETS:
1. I can use limits to determine the continuity of a function.
1. I can apply the Intermediate Value Theorem to
continuous functions.
1. I can use limits to describe end behavior.
Continuity, End Behavior, and Limits
 The graph of a _____________ ____________
has no breaks, holes, or gaps.

Simple Check: Can you trace the graph without lifting your
pencil? If you can, it is continuous.
 For a function f(x) to be continuous at x = c, the
function must approach a unique function value as
the x-values approach c from the left and right sides.
 The concept of approaching a value without
necessarily ever reaching it is called a _________.
Continuity, End Behavior, and Limits
If the value of f(x) approaches a specific value L when x
approaches c from BOTH sides, then the limit of f(x)
as x approaches c is L.
lim f (x)  L
x c
“The limit of f(x) as x
approaches c is L.”
Continuity, End Behavior, and Limits
 Functions that are NOT continuous are called
_______________ functions.
 Three types of Discontinuity:

1. _____________ Discontinuity

2. _____________ Discontinuity

3. _____________ Discontinuity
Infinite Discontinuity
 A function has infinite discontinuity at x = c if the
function value increases or decreases indefinitely (to
infinity) as x approaches c from the left and right.
Jump Discontinuity
 A function has a jump discontinuity at x = c if the
limits of the function as x approaches c from the left
and right exist but have two distinct values.
Removable Discontinuity
 A function has a removable discontinuity if the function
is continuous everywhere except for a hole at x = c.
Although the limit exists at c, the function is not defined at
c, so it’s not continuous.
Nonremovable Discontinuities
 Infinite and jump discontinuities are classified as
nonremovable discontinuities. A nonremovable
discontinuity cannot be eliminated by redefining the
function at that point.
Continuity Test
 A function f(x) is continuous at x = c if it satisfies the
following conditions:

1. f(x) is defined at c. (In short—f(c) exists)

2. f(x) approaches the same value from either side of c. (In
short—the limit of f(x) as x approaches c exists)

3. The value that f(x) approaches from each side of c is f(c). (In
short—the limit of f(x) as x approaches c is f(c).
End Behavior
 The end behavior of a function describes how a
function behaves at either end of the graph.

In other words, end behavior is what happens to the value of
f(x) as x increases or decreases without bound (either
becoming larger and larger or more and more negative).
 We describe the end behavior of a graph using the
concept of a limit.
Continuity, End Behavior, and Limits
1. Is f (x)  1
2x 1
continuous at x = ½? Justify using
the continuity test.


Continuity, End Behavior, and Limits
2. Is f (x)  x 3 continuous at x = 0? Justify using
the continuity test.

Continuity, End Behavior, and Limits
3. Determine whether the function is continuous at the
given x-value. Justify using the continuity test. If it
is discontinuous, identify the type of discontinuity.
1
at x = 1
f (x) 
x 1

Continuity, End Behavior, and Limits
4. Determine whether the function is continuous at the
given x-value. Justify using the continuity test. If it
is discontinuous, identify the type of discontinuity.
x  2 at x = 2
f (x)  2
x 4
Is there a zero between 0 and 10?
Intermediate Value Theorem
 If f(x) is a continuous function and a < b and there is
a value n such that n is between f(a) and f(b), then
there is a number c, such that a < c < b and f(c) = n.
Yes
Corollary: The Location Principle
 If f(x) is a continuous function and f(a) and f(b) have
opposite signs, then there exists at least one value c
such that a < c < b and f(c) = 0. That is, there is a
zero between a and b.
Continuity, End Behavior, and Limits
5. Determine between which consecutive integers the
real zeros of each function are located on the given
interval.
3
2
a. f (x)  x  x  [-2, 2]
4
b. f (x)  x 3  2x  5 [-2, 2]
Continuity, End Behavior, and Limits
6. Use the graph of f (x)  x 3  x 2  4x  4 to describe the
end behavior of the function. Support your
conjecture numerically.

Continuity, End Behavior, and Limits
x 2
f (x)  2
x  x  2 to describe the
7. Use the graph of
end behavior of the function. Support your
conjecture numerically.

Homework
Pages 30-33; Problems 1, 3, 7, 9, 13, 15, 19, 21, 23, 25,
33, 35, 37, 39, 47, 50, 55