Chapter 11
Introduction to
Calculus
11.3 Limits and Continuity
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1
Objectives:
•
•
Determine whether a function is continuous at a
number.
Determine for what numbers a function is
discontinuous.
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Definition of a Function Continuous at a Number
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Example: Determining Whether a Function is
Continuous at a Number
Determine whether the following function is continuous at 1:
x2
f ( x)  2
x 4
According to the definition, three conditions must be satisfied to have
continuity at a.
Condition 1 f is defined at a.
1  2 1 1
f (1)  2


1  4 3 3
Because f(1) is a real number, f(1) is defined.
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Example: Determining Whether a Function is
Continuous at a Number
Determine whether the following function is continuous at 1:
x2
f ( x)  2
x 4
According to the definition, three conditions must be satisfied to have
continuity at a. Condition 1, that f is defined at a, has been satisfied
Condition 2 lim f ( x ) exists.
x a
lim( x  2)
x2
1 2
1 1 1
x 1

lim f ( x)  lim 2
 2



2
x 1
x1 x  4
lim  x  4  1  4 1  4 3 3
x 1
x2
Using properties of limits, we see that lim 2
x 1 x  4
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exists.
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Example: Determining Whether a Function is
Continuous at a Number
Determine whether the following function is continuous at 1:
x2
f ( x)  2
x 4
According to the definition, three conditions must be satisfied to have
continuity at a. Conditions 1 and 2 have been satisfied.
Condition 3
x2 1
lim 2

x 1 x  4
3
1
f (1) 
3
lim f ( x )  f (a )
x a
Because the three conditions are
satisfied, we conclude that f is
continuous at 1.
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Example: Determining Whether a Function is
Continuous at a Number
Determine whether the following function is continuous at 2:
x2
f ( x)  2
x 4
According to the definition, three conditions must be satisfied to have
continuity at a.
Condition 1 f is defined at a.
Factor the denominator of the function’s equation:
x2
x2
f ( x)  2

x  4 ( x  2)( x  2)
f is not defined at 2.
Therefore, f is not
continuous at 2. We say
“f is discontinuous at 2”.
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Determining Where Functions are Discontinuous
If f is a polynomial function, lim f ( x)  f (a ) for any
x a
number a. A polynomial function is continuous at every
number. Many functions are continuous at every number
in their domain. Rational, exponential, logarithmic, sine,
cosine, tangent, cotangent, secant, and cosecant functions
are continuous at every number in their respective domains.
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Example: Determining Where a Piecewise Function is
Discontinuous
Determine for what numbers x, if any, the following
function is discontinuous:
if x  0
 2x
 2
f ( x)   x  1 if 0  x  2
7  x
if x  2

First, we determine whether each of the three pieces of f is continuous.
The first piece, a linear function, is continuous at every number x.
The second piece, a quadratic function, is continuous at every number x.
The third piece, a linear function, is continuous at every number x.
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Example: Determining Where a Piecewise Function is
Discontinuous
Determine for what numbers x, if any, the following
function is discontinuous:
if x  0
 2x
 2
f ( x)   x  1 if 0  x  2
7  x
if x  2

We have determined that each of the three pieces of the function are
continuous at x. We now investigate continuity at x = 0 and x = 2.
We begin by investigating continuity at x = 0.
Condition 1 f is defined at a.
Because f(0) is a real number,
f (0)  2(0)  0
f(0) is defined.
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Example: Determining Where a Piecewise Function is
Discontinuous
Determine for what numbers x, if any, the following function is
discontinuous:
if x  0
 2x
 2
f ( x)   x  1 if 0  x  2
7  x
if x  2

We are investigating the continuity of f at x = 0.
Condition 2 lim f ( x ) exists.
The left- and right-hand
limits are not equal. This

2(0)

0
lim f ( x)  lim 2 x
means that the limit does
x 0
x 0
not exist. The function
2
2
lim f ( x)  lim  x  1  1  1  2
is discontinuous at x = 0.
x 0
x 0
x a
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