Defining Limits One-Sided Limits Infinite Limits
Lecture 1: Introduction to Limits
Defining LimitsWhy Limits?Example 1 – An Indeterminate FormDefinition of LimitExample 2 – Applying the Definition
One-Sided LimitsDoes a Limit Always Exist?Example 3 – A Step FunctionDefinition of One-Sided LimitsChecking for the Existence of a LimitExample 4 – Calculating Limits
Infinite LimitsDefinition of Infinite LimitsExample 5 – Evaluating Infinite Limits
Clint Lee Math 112 Lecture 1: Introduction to Limits 1/22
Defining Limits One-Sided Limits Infinite Limits
Why Limits?
Why Limits?
Limits are used in calculus to
Clint Lee Math 112 Lecture 1: Introduction to Limits 2/22
Defining Limits One-Sided Limits Infinite Limits
Why Limits?
Why Limits?
Limits are used in calculus to
define and calculate the slope of a tangent line
Clint Lee Math 112 Lecture 1: Introduction to Limits 2/22
Defining Limits One-Sided Limits Infinite Limits
Why Limits?
Why Limits?
Limits are used in calculus to
define and calculate the slope of a tangent line
calculate velocities and rates of change
Clint Lee Math 112 Lecture 1: Introduction to Limits 2/22
Defining Limits One-Sided Limits Infinite Limits
Why Limits?
Why Limits?
Limits are used in calculus to
define and calculate the slope of a tangent line
calculate velocities and rates of change
define and calculate areas and volumes
Clint Lee Math 112 Lecture 1: Introduction to Limits 2/22
Defining Limits One-Sided Limits Infinite Limits
Why Limits?
Why Limits?
Limits are used in calculus to
define and calculate the slope of a tangent line
calculate velocities and rates of change
define and calculate areas and volumes
But, more generally, limits provide a way to extend the operation offunction evaluation.
Clint Lee Math 112 Lecture 1: Introduction to Limits 2/22
Defining Limits One-Sided Limits Infinite Limits
Example 1 – An Indeterminate Form
Example 1 – An Indeterminate Form
Consider the function
f (x) =x + 1
x2 − 1
The domain of this function is
Clint Lee Math 112 Lecture 1: Introduction to Limits 3/22
Defining Limits One-Sided Limits Infinite Limits
Example 1 – An Indeterminate Form
Example 1 – An Indeterminate Form
Consider the function
f (x) =x + 1
x2 − 1
The domain of this function is{
x∣
∣ − ∞ < x < −1 or − 1 < x < 1 or 1 < x < ∞
}
=
Clint Lee Math 112 Lecture 1: Introduction to Limits 3/22
Defining Limits One-Sided Limits Infinite Limits
Example 1 – An Indeterminate Form
Example 1 – An Indeterminate Form
Consider the function
f (x) =x + 1
x2 − 1
The domain of this function is{
x∣
∣ − ∞ < x < −1 or − 1 < x < 1 or 1 < x < ∞
}
=
(−∞,−1) ∪ (−1, 1) ∪ (1, ∞)
Clint Lee Math 112 Lecture 1: Introduction to Limits 3/22
Defining Limits One-Sided Limits Infinite Limits
Example 1 – An Indeterminate Form
Example 1 – An Indeterminate Form
Consider the function
f (x) =x + 1
x2 − 1
The domain of this function is{
x∣
∣ − ∞ < x < −1 or − 1 < x < 1 or 1 < x < ∞
}
=
(−∞,−1) ∪ (−1, 1) ∪ (1, ∞)
That is the function f is defined except at x = 1 and x = −1. Let’s seewhat happens when we evaluate f at these points.
Clint Lee Math 112 Lecture 1: Introduction to Limits 3/22
Defining Limits One-Sided Limits Infinite Limits
Example 1 – An Indeterminate Form
Continuing Example 1
Evaluating f at x = 1 gives
f (1) =2
0
which is obviously undefined. Use yourgraphing calculator or Maple to plot thegraph of this function in the viewingrectangle
[
−2, 3]
×[
−4, 4]
.
Clint Lee Math 112 Lecture 1: Introduction to Limits 4/22
Defining Limits One-Sided Limits Infinite Limits
Example 1 – An Indeterminate Form
Continuing Example 1
Evaluating f at x = 1 gives
f (1) =2
0
which is obviously undefined. Use yourgraphing calculator or Maple to plot thegraph of this function in the viewingrectangle
[
−2, 3]
×[
−4, 4]
.
1
Does your graph look like this?
Clint Lee Math 112 Lecture 1: Introduction to Limits 4/22
Defining Limits One-Sided Limits Infinite Limits
Example 1 – An Indeterminate Form
Continuing Example 1
Evaluating f at x = 1 gives
f (1) =2
0
which is obviously undefined. Use yourgraphing calculator or Maple to plot thegraph of this function in the viewingrectangle
[
−2, 3]
×[
−4, 4]
.
1
Does your graph look like this? What’s wrong with this picture?
Clint Lee Math 112 Lecture 1: Introduction to Limits 4/22
Defining Limits One-Sided Limits Infinite Limits
Example 1 – An Indeterminate Form
Continuing Example 1
Evaluating f at x = 1 gives
f (1) =2
0
which is obviously undefined. Use yourgraphing calculator or Maple to plot thegraph of this function in the viewingrectangle
[
−2, 3]
×[
−4, 4]
.
1
The almost vertical line through x = 1 is an artifact. The function is notdefined at x = 1, or at x = −1.
Clint Lee Math 112 Lecture 1: Introduction to Limits 4/22
Defining Limits One-Sided Limits Infinite Limits
Example 1 – An Indeterminate Form
Continuing Example 1
Evaluating f at x = 1 gives
f (1) =2
0
which is obviously undefined. Use yourgraphing calculator or Maple to plot thegraph of this function in the viewingrectangle
[
−2, 3]
×[
−4, 4]
.
x=
1
Here is the correct graph with a vertical asymptote at x = 1 and an opencircle at x = −1.
Clint Lee Math 112 Lecture 1: Introduction to Limits 4/22
Defining Limits One-Sided Limits Infinite Limits
Example 1 – An Indeterminate Form
Continuing Example 1
Evaluating f at x = −1 gives
f (−1) =0
0
which is again undefined. However, now we do not necessarily getan arbitrarily large value, as in the last case.
Clint Lee Math 112 Lecture 1: Introduction to Limits 5/22
Defining Limits One-Sided Limits Infinite Limits
Example 1 – An Indeterminate Form
Continuing Example 1
Evaluating f at x = −1 gives
f (−1) =0
0
which is again undefined. However, now we do not necessarily getan arbitrarily large value, as in the last case. This is an indeterminateform.
Clint Lee Math 112 Lecture 1: Introduction to Limits 5/22
Defining Limits One-Sided Limits Infinite Limits
Example 1 – An Indeterminate Form
Continuing Example 1
Evaluating f at x = −1 gives
f (−1) =0
0
which is again undefined. However, now we do not necessarily getan arbitrarily large value, as in the last case. This is an indeterminateform. Something different happens at x = −1 than at x = 1.
Clint Lee Math 112 Lecture 1: Introduction to Limits 5/22
Defining Limits One-Sided Limits Infinite Limits
Example 1 – An Indeterminate Form
Continuing Example 1
Evaluating f at x = −1 gives
f (−1) =0
0
which is again undefined. However, now we do not necessarily getan arbitrarily large value, as in the last case. This is an indeterminateform. Something different happens at x = −1 than at x = 1. Thedifference is hinted at by the graph we saw earlier.
Clint Lee Math 112 Lecture 1: Introduction to Limits 5/22
Defining Limits One-Sided Limits Infinite Limits
Example 1 – An Indeterminate Form
Continuing Example 1
Evaluating f at x = −1 gives
f (−1) =0
0
which is again undefined. However, now we do not necessarily getan arbitrarily large value, as in the last case. This is an indeterminateform. Something different happens at x = −1 than at x = 1. Thedifference is hinted at by the graph we saw earlier. There is an opencircle on the graph at x = −1.
Clint Lee Math 112 Lecture 1: Introduction to Limits 5/22
Defining Limits One-Sided Limits Infinite Limits
Example 1 – An Indeterminate Form
Continuing Example 1
To investigate what happens to fat x = −1 make a table of valuesnear x = −1.
x f (x)
Clint Lee Math 112 Lecture 1: Introduction to Limits 6/22
Defining Limits One-Sided Limits Infinite Limits
Example 1 – An Indeterminate Form
Continuing Example 1
To investigate what happens to fat x = −1 make a table of valuesnear x = −1.
x f (x)
−2.000 −0.333333
0.000 −1.000000
Clint Lee Math 112 Lecture 1: Introduction to Limits 6/22
Defining Limits One-Sided Limits Infinite Limits
Example 1 – An Indeterminate Form
Continuing Example 1
To investigate what happens to fat x = −1 make a table of valuesnear x = −1.
x f (x)
−2.000 −0.333333
−1.500 −0.400000
−0.500 −0.666667
0.000 −1.000000
Clint Lee Math 112 Lecture 1: Introduction to Limits 6/22
Defining Limits One-Sided Limits Infinite Limits
Example 1 – An Indeterminate Form
Continuing Example 1
To investigate what happens to fat x = −1 make a table of valuesnear x = −1.
x f (x)
−2.000 −0.333333
−1.500 −0.400000
−1.100 −0.476190
−0.900 −0.526316
−0.500 −0.666667
0.000 −1.000000
Clint Lee Math 112 Lecture 1: Introduction to Limits 6/22
Defining Limits One-Sided Limits Infinite Limits
Example 1 – An Indeterminate Form
Continuing Example 1
To investigate what happens to fat x = −1 make a table of valuesnear x = −1.
x f (x)
−2.000 −0.333333
−1.500 −0.400000
−1.100 −0.476190
−1.010 −0.497512
−0.990 −0.502513
−0.900 −0.526316
−0.500 −0.666667
0.000 −1.000000
Clint Lee Math 112 Lecture 1: Introduction to Limits 6/22
Defining Limits One-Sided Limits Infinite Limits
Example 1 – An Indeterminate Form
Continuing Example 1
To investigate what happens to fat x = −1 make a table of valuesnear x = −1.
x f (x)
−2.000 −0.333333
−1.500 −0.400000
−1.100 −0.476190
−1.010 −0.497512
−1.001 −0.499750
−0.999 −0.500250
−0.990 −0.502513
−0.900 −0.526316
−0.500 −0.666667
0.000 −1.000000
Clint Lee Math 112 Lecture 1: Introduction to Limits 6/22
Defining Limits One-Sided Limits Infinite Limits
Example 1 – An Indeterminate Form
Continuing Example 1
To investigate what happens to fat x = −1 make a table of valuesnear x = −1.
It appears that the value of f getsclose to
−0.5 = −1
2
as x gets close to −1.
x f (x)
−2.000 −0.333333
−1.500 −0.400000
−1.100 −0.476190
−1.010 −0.497512
−1.001 −0.499750
−0.999 −0.500250
−0.990 −0.502513
−0.900 −0.526316
−0.500 −0.666667
0.000 −1.000000
Clint Lee Math 112 Lecture 1: Introduction to Limits 6/22
Defining Limits One-Sided Limits Infinite Limits
Definition of Limit
Definition of Limit
From Example 1 we see that even though f (−1) is not defined there isa finite and precisely defined limiting value for f at x = −1.
Clint Lee Math 112 Lecture 1: Introduction to Limits 7/22
Defining Limits One-Sided Limits Infinite Limits
Definition of Limit
Definition of Limit
From Example 1 we see that even though f (−1) is not defined there isa finite and precisely defined limiting value for f at x = −1. We saythat
limx→−1
x + 1
x2 − 1= −
1
2
The following definition makes this more precise.
Clint Lee Math 112 Lecture 1: Introduction to Limits 7/22
Defining Limits One-Sided Limits Infinite Limits
Definition of Limit
Definition of Limit
From Example 1 we see that even though f (−1) is not defined there isa finite and precisely defined limiting value for f at x = −1. We saythat
limx→−1
x + 1
x2 − 1= −
1
2
The following definition makes this more precise.
Definition (Limit of a Function)
We say thatlimx→a
f (x) = L
if f (x) can be made arbitrarily close to L by making x sufficiently closeto a, on either side of a.
Clint Lee Math 112 Lecture 1: Introduction to Limits 7/22
Defining Limits One-Sided Limits Infinite Limits
Example 2 – Applying the Definition
Example 2 – Applying the Definition
To apply the definition, suppose thatwe want to make sure that the valueof f (x) is within 0.2 of limit valueL = −0.5. This means that we want
−0.7 ≤ f (x) ≤ −0.3
Clint Lee Math 112 Lecture 1: Introduction to Limits 8/22
Defining Limits One-Sided Limits Infinite Limits
Example 2 – Applying the Definition
Example 2 – Applying the Definition
To apply the definition, suppose thatwe want to make sure that the valueof f (x) is within 0.2 of limit valueL = −0.5. This means that we want
−0.7 ≤ f (x) ≤ −0.3
From the table of values in Example 1 we see that
f (−1.5) = −0.40000 < −0.3 and f (−0.5) = −0.666667 > −0.7
Clint Lee Math 112 Lecture 1: Introduction to Limits 8/22
Defining Limits One-Sided Limits Infinite Limits
Example 2 – Applying the Definition
Example 2 – Applying the Definition
To apply the definition, suppose thatwe want to make sure that the valueof f (x) is within 0.2 of limit valueL = −0.5. This means that we want
−0.7 ≤ f (x) ≤ −0.3
From the table of values in Example 1 we see that
f (−1.5) = −0.40000 < −0.3 and f (−0.5) = −0.666667 > −0.7
Hence, we only need to have x within 0.5 of the limit point x = −1, sothat
− 1.5 ≤ x ≤ −0.5
Clint Lee Math 112 Lecture 1: Introduction to Limits 8/22
Defining Limits One-Sided Limits Infinite Limits
Example 2 – Applying the Definition
Example 2 – Applying the Definition
To apply the definition, suppose thatwe want to make sure that the valueof f (x) is within 0.2 of limit valueL = −0.5. This means that we want
−0.7 ≤ f (x) ≤ −0.3
From the table of values in Example 1 we see that
f (−1.5) = −0.40000 < −0.3 and f (−0.5) = −0.666667 > −0.7
Hence, we only need to have x within 0.5 of the limit point x = −1, sothat
− 1.5 ≤ x ≤ −0.5
Hence for x in the interval [−1.5,−0.5] the values of f (x) are in theinterval [−0.6666667,−0.4], which is contained in the interval[−0.7,−0.3].
Clint Lee Math 112 Lecture 1: Introduction to Limits 8/22
Defining Limits One-Sided Limits Infinite Limits
Example 2 – Applying the Definition
Example 2 – Applying the Definition
To apply the definition, suppose thatwe want to make sure that the valueof f (x) is within 0.2 of limit valueL = −0.5. This means that we want
−0.7 ≤ f (x) ≤ −0.3
The graph shows the relationbetween these two intervals.
−1.5 −0.5
−0.3
−0.7
From the table of values in Example 1 we see that
f (−1.5) = −0.40000 < −0.3 and f (−0.5) = −0.666667 > −0.7
Hence, we only need to have x within 0.5 of the limit point x = −1, sothat
− 1.5 ≤ x ≤ −0.5
Hence for x in the interval [−1.5,−0.5] the values of f (x) are in theinterval [−0.6666667,−0.4], which is contained in the interval[−0.7,−0.3]. The graph of the function f lies entirely in the boxbounded by the horizontal lines and the vertical lines.
Clint Lee Math 112 Lecture 1: Introduction to Limits 8/22
Defining Limits One-Sided Limits Infinite Limits
Example 2 – Applying the Definition
Continuing Example 2
If we decrease the vertical extent of the box, which means we want tokeep the values of f (x) closer to the limit value, we must decrease itshorizontal extent as well. From the graph it appears that no matterhow small we make the vertical extent of the box, we can make thebox narrow enough (horizontally) so that the graph of f stays entirelyinside the box.
Clint Lee Math 112 Lecture 1: Introduction to Limits 9/22
Defining Limits One-Sided Limits Infinite Limits
Example 2 – Applying the Definition
Continuing Example 2
If we decrease the vertical extent of the box, which means we want tokeep the values of f (x) closer to the limit value, we must decrease itshorizontal extent as well. From the graph it appears that no matterhow small we make the vertical extent of the box, we can make thebox narrow enough (horizontally) so that the graph of f stays entirelyinside the box.To see another step in the process of further reducing the verticalextent, width, of the box click on the button below.
Clint Lee Math 112 Lecture 1: Introduction to Limits 9/22
Defining Limits One-Sided Limits Infinite Limits
Example 2 – Applying the Definition
Continuing Example 2
If we decrease the vertical extent of the box, which means we want tokeep the values of f (x) closer to the limit value, we must decrease itshorizontal extent as well. From the graph it appears that no matterhow small we make the vertical extent of the box, we can make thebox narrow enough (horizontally) so that the graph of f stays entirelyinside the box.To see another step in the process of further reducing the verticalextent, width, of the box click on the button below.
Reducing width of box
Clint Lee Math 112 Lecture 1: Introduction to Limits 9/22
Defining Limits One-Sided Limits Infinite Limits
Does a Limit Always Exist?
Does a Limit Always Exist?
An important feature of the definition of the limit of a function is
Clint Lee Math 112 Lecture 1: Introduction to Limits 10/22
Defining Limits One-Sided Limits Infinite Limits
Does a Limit Always Exist?
Does a Limit Always Exist?
An important feature of the definition of the limit of a function is
Two Sided Nature of Limits
We must be able to make f (x) arbitrarily close to L by making xsufficiently close to a, on either side of a.
Clint Lee Math 112 Lecture 1: Introduction to Limits 10/22
Defining Limits One-Sided Limits Infinite Limits
Does a Limit Always Exist?
Does a Limit Always Exist?
An important feature of the definition of the limit of a function is
Two Sided Nature of Limits
We must be able to make f (x) arbitrarily close to L by making xsufficiently close to a, on either side of a.
In some cases we get a different value if we make x close to a ondifferent sides of a.
Clint Lee Math 112 Lecture 1: Introduction to Limits 10/22
Defining Limits One-Sided Limits Infinite Limits
Does a Limit Always Exist?
Does a Limit Always Exist?
An important feature of the definition of the limit of a function is
Two Sided Nature of Limits
We must be able to make f (x) arbitrarily close to L by making xsufficiently close to a, on either side of a.
In some cases we get a different value if we make x close to a ondifferent sides of a. In this case the limit does not exist.
Clint Lee Math 112 Lecture 1: Introduction to Limits 10/22
Defining Limits One-Sided Limits Infinite Limits
Does a Limit Always Exist?
Does a Limit Always Exist?
An important feature of the definition of the limit of a function is
Two Sided Nature of Limits
We must be able to make f (x) arbitrarily close to L by making xsufficiently close to a, on either side of a.
In some cases we get a different value if we make x close to a ondifferent sides of a. In this case the limit does not exist.This leads to the idea of one-sided limits.
Clint Lee Math 112 Lecture 1: Introduction to Limits 10/22
Defining Limits One-Sided Limits Infinite Limits
Does a Limit Always Exist?
Does a Limit Always Exist?
An important feature of the definition of the limit of a function is
Two Sided Nature of Limits
We must be able to make f (x) arbitrarily close to L by making xsufficiently close to a, on either side of a.
In some cases we get a different value if we make x close to a ondifferent sides of a. In this case the limit does not exist.This leads to the idea of one-sided limits. We will introduce afunction in which one-sided limits play a role in Example 3 and thengive the definition of one-sided limits.
Clint Lee Math 112 Lecture 1: Introduction to Limits 10/22
Defining Limits One-Sided Limits Infinite Limits
Does a Limit Always Exist?
Does a Limit Always Exist?
An important feature of the definition of the limit of a function is
Two Sided Nature of Limits
We must be able to make f (x) arbitrarily close to L by making xsufficiently close to a, on either side of a.
In some cases we get a different value if we make x close to a ondifferent sides of a. In this case the limit does not exist.This leads to the idea of one-sided limits. We will introduce afunction in which one-sided limits play a role in Example 3 and thengive the definition of one-sided limits.There are other cases in which a limit does not exist. We willinvestigate some of them later.
Clint Lee Math 112 Lecture 1: Introduction to Limits 10/22
Defining Limits One-Sided Limits Infinite Limits
Example 3 – A Step Function
Example 3 – A Step Function
Consider the function
g(x) =|x|
x
−1
1
x
y
Clint Lee Math 112 Lecture 1: Introduction to Limits 11/22
Defining Limits One-Sided Limits Infinite Limits
Example 3 – A Step Function
Example 3 – A Step Function
Consider the function
g(x) =|x|
x=
{
−1 if x < 01 if x > 0
This is a piecewise function forwhich the formula for the functionis different on different intervals.
−1
1
x
y
Clint Lee Math 112 Lecture 1: Introduction to Limits 11/22
Defining Limits One-Sided Limits Infinite Limits
Example 3 – A Step Function
Example 3 – A Step Function
Consider the function
g(x) =|x|
x=
{
−1 if x < 01 if x > 0
This is a piecewise function forwhich the formula for the functionis different on different intervals.Its graph looks like this:
−1
1
x
y
Clint Lee Math 112 Lecture 1: Introduction to Limits 11/22
Defining Limits One-Sided Limits Infinite Limits
Example 3 – A Step Function
Example 3 – A Step Function
Consider the function
g(x) =|x|
x=
{
−1 if x < 01 if x > 0
This is a piecewise function forwhich the formula for the functionis different on different intervals.Its graph looks like this:
−1
1
x
y
Clint Lee Math 112 Lecture 1: Introduction to Limits 11/22
Defining Limits One-Sided Limits Infinite Limits
Example 3 – A Step Function
Example 3 – A Step Function
Consider the function
g(x) =|x|
x=
{
−1 if x < 01 if x > 0
This is a piecewise function forwhich the formula for the functionis different on different intervals.Its graph looks like this:
−1
1
x
y
This particular piecewise functionis called a step function.
Clint Lee Math 112 Lecture 1: Introduction to Limits 11/22
Defining Limits One-Sided Limits Infinite Limits
Example 3 – A Step Function
The Absolute Value Function (A digression)
The absolute value function isanother piecewise function. Itsformula is
x
y
Clint Lee Math 112 Lecture 1: Introduction to Limits 12/22
Defining Limits One-Sided Limits Infinite Limits
Example 3 – A Step Function
The Absolute Value Function (A digression)
The absolute value function isanother piecewise function. Itsformula is
|x| =
−x if x < 00 if x = 0x if x > 0
x
y
Clint Lee Math 112 Lecture 1: Introduction to Limits 12/22
Defining Limits One-Sided Limits Infinite Limits
Example 3 – A Step Function
The Absolute Value Function (A digression)
The absolute value function isanother piecewise function. Itsformula is
|x| =
−x if x < 00 if x = 0x if x > 0
Its graph looks like this:
x
y
Clint Lee Math 112 Lecture 1: Introduction to Limits 12/22
Defining Limits One-Sided Limits Infinite Limits
Example 3 – A Step Function
The Absolute Value Function (A digression)
The absolute value function isanother piecewise function. Itsformula is
|x| =
−x if x < 00 if x = 0x if x > 0
Its graph looks like this:
x
y
There is no jump in the graph ofthis piecewise function. Not allpiecewise functions have a jumpin their graphs.
Clint Lee Math 112 Lecture 1: Introduction to Limits 12/22
Defining Limits One-Sided Limits Infinite Limits
Example 3 – A Step Function
Continuing Example 3
For the function g defined at the beginning of Example 3, what canwe say about the limit
limx→0
g(x)?
Clint Lee Math 112 Lecture 1: Introduction to Limits 13/22
Defining Limits One-Sided Limits Infinite Limits
Example 3 – A Step Function
Continuing Example 3
For the function g defined at the beginning of Example 3, what canwe say about the limit
limx→0
g(x)?
We must say that the limit does not exist. This is because we cannotmake g(x) close to a single value as x gets close to x = 0.
Clint Lee Math 112 Lecture 1: Introduction to Limits 13/22
Defining Limits One-Sided Limits Infinite Limits
Example 3 – A Step Function
Continuing Example 3
For the function g defined at the beginning of Example 3, what canwe say about the limit
limx→0
g(x)?
We must say that the limit does not exist. This is because we cannotmake g(x) close to a single value as x gets close to x = 0. In fact, forall x < 0, g(x) = −1, and for all x > 0, g(x) = 1.
Clint Lee Math 112 Lecture 1: Introduction to Limits 13/22
Defining Limits One-Sided Limits Infinite Limits
Example 3 – A Step Function
Continuing Example 3
For the function g defined at the beginning of Example 3, what canwe say about the limit
limx→0
g(x)?
We must say that the limit does not exist. This is because we cannotmake g(x) close to a single value as x gets close to x = 0. In fact, forall x < 0, g(x) = −1, and for all x > 0, g(x) = 1.However, we can write
limx→0−
g(x) = −1 and limx→0+
g(x) = 1
These are one-sided limits.
Clint Lee Math 112 Lecture 1: Introduction to Limits 13/22
Defining Limits One-Sided Limits Infinite Limits
Definition of One-Sided Limits
Definition of One-Sided Limits
As seen in Example 3 a function can get close to different values asthe limit point a is approached from the two different sides. Thisleads to:
Clint Lee Math 112 Lecture 1: Introduction to Limits 14/22
Defining Limits One-Sided Limits Infinite Limits
Definition of One-Sided Limits
Definition of One-Sided Limits
As seen in Example 3 a function can get close to different values asthe limit point a is approached from the two different sides. Thisleads to:
Definition (Limits from the Right and Left)
Clint Lee Math 112 Lecture 1: Introduction to Limits 14/22
Defining Limits One-Sided Limits Infinite Limits
Definition of One-Sided Limits
Definition of One-Sided Limits
As seen in Example 3 a function can get close to different values asthe limit point a is approached from the two different sides. Thisleads to:
Definition (Limits from the Right and Left)
We say thatlim
x→a+f (x) = R
if f (x) can be made arbitrarily close to R by making x sufficientlyclose to a with x > a, that is for x to the right of a. This is the limit of fat a from the right, or from above,
Clint Lee Math 112 Lecture 1: Introduction to Limits 14/22
Defining Limits One-Sided Limits Infinite Limits
Definition of One-Sided Limits
Definition of One-Sided Limits
As seen in Example 3 a function can get close to different values asthe limit point a is approached from the two different sides. Thisleads to:
Definition (Limits from the Right and Left)
We say thatlim
x→a+f (x) = R
if f (x) can be made arbitrarily close to R by making x sufficientlyclose to a with x > a, that is for x to the right of a. This is the limit of fat a from the right, or from above, and
limx→a−
f (x) = L
if f (x) can be made arbitrarily close to L by making x sufficiently closeto a with x < a, that is for x to the left of a. This is the limit of f at afrom the left, or from below.
Clint Lee Math 112 Lecture 1: Introduction to Limits 14/22
Defining Limits One-Sided Limits Infinite Limits
Checking for the Existence of a Limit
Checking for the Existence of a Limit
We can check to see if a limit exists by checking the one-sided limits.
Clint Lee Math 112 Lecture 1: Introduction to Limits 15/22
Defining Limits One-Sided Limits Infinite Limits
Checking for the Existence of a Limit
Checking for the Existence of a Limit
We can check to see if a limit exists by checking the one-sided limits.
Existence of a Limit
Clint Lee Math 112 Lecture 1: Introduction to Limits 15/22
Defining Limits One-Sided Limits Infinite Limits
Checking for the Existence of a Limit
Checking for the Existence of a Limit
We can check to see if a limit exists by checking the one-sided limits.
Existence of a Limit
The limitlimx→a
f (x)
exists and is equal to L if and only if both
Clint Lee Math 112 Lecture 1: Introduction to Limits 15/22
Defining Limits One-Sided Limits Infinite Limits
Checking for the Existence of a Limit
Checking for the Existence of a Limit
We can check to see if a limit exists by checking the one-sided limits.
Existence of a Limit
The limitlimx→a
f (x)
exists and is equal to L if and only if both
limx→a−
f (x) and limx→a+
f (x)
exist, and
Clint Lee Math 112 Lecture 1: Introduction to Limits 15/22
Defining Limits One-Sided Limits Infinite Limits
Checking for the Existence of a Limit
Checking for the Existence of a Limit
We can check to see if a limit exists by checking the one-sided limits.
Existence of a Limit
The limitlimx→a
f (x)
exists and is equal to L if and only if both
limx→a−
f (x) and limx→a+
f (x)
exist, andlim
x→a−f (x) = L and lim
x→a+f (x) = L
Clint Lee Math 112 Lecture 1: Introduction to Limits 15/22
Defining Limits One-Sided Limits Infinite Limits
Checking for the Existence of a Limit
Checking for the Existence of a Limit
We can check to see if a limit exists by checking the one-sided limits.
Existence of a Limit
The limitlimx→a
f (x)
exists and is equal to L if and only if both
limx→a−
f (x) and limx→a+
f (x)
exist, andlim
x→a−f (x) = L and lim
x→a+f (x) = L
Otherwise, the regular, two-sided, limit does not exist.
Clint Lee Math 112 Lecture 1: Introduction to Limits 15/22
Defining Limits One-Sided Limits Infinite Limits
Example 4 – Calculating Limits
Example 4 – Calculating Limits
Consider the function
h(x) =
2x + 8 if x < −1−2x + 4 if −1 ≤ x < 1
x2 − 2x + 2 if x ≥ 1
Graph each branch separately:2
4
6
2−2x
y
Clint Lee Math 112 Lecture 1: Introduction to Limits 16/22
Defining Limits One-Sided Limits Infinite Limits
Example 4 – Calculating Limits
Example 4 – Calculating Limits
Consider the function
h(x) =
2x + 8 if x < −1−2x + 4 if −1 ≤ x < 1
x2 − 2x + 2 if x ≥ 1
Graph each branch separately:
y = 2x + 8
line thru (−3, 2) and (−1, 6)
2
4
6
2−2x
y
Clint Lee Math 112 Lecture 1: Introduction to Limits 16/22
Defining Limits One-Sided Limits Infinite Limits
Example 4 – Calculating Limits
Example 4 – Calculating Limits
Consider the function
h(x) =
2x + 8 if x < −1−2x + 4 if −1 ≤ x < 1
x2 − 2x + 2 if x ≥ 1
Graph each branch separately:
y = 2x + 8
line thru (−3, 2) and (−1, 6)
y = −2x + 4
line thru (−1, 6) and (1, 2)
2
4
6
2−2x
y
Clint Lee Math 112 Lecture 1: Introduction to Limits 16/22
Defining Limits One-Sided Limits Infinite Limits
Example 4 – Calculating Limits
Example 4 – Calculating Limits
Consider the function
h(x) =
2x + 8 if x < −1−2x + 4 if −1 ≤ x < 1
x2 − 2x + 2 if x ≥ 1
Graph each branch separately:
y = 2x + 8
line thru (−3, 2) and (−1, 6)
y = −2x + 4
line thru (−1, 6) and (1, 2)
y = x2 − 2x + 2 = (x − 1)2 + 1
parabola
2
4
6
2−2x
y
Clint Lee Math 112 Lecture 1: Introduction to Limits 16/22
Defining Limits One-Sided Limits Infinite Limits
Example 4 – Calculating Limits
Example 4 – Calculating Limits
Consider the function
h(x) =
2x + 8 if x < −1−2x + 4 if −1 ≤ x < 1
x2 − 2x + 2 if x ≥ 1
Graph each branch separately:
y = 2x + 8
line thru (−3, 2) and (−1, 6)
y = −2x + 4
line thru (−1, 6) and (1, 2)
y = x2 − 2x + 2 = (x − 1)2 + 1
parabola
2
4
6
2−2x
y
This is a piecewise functionwith a jump at x = 1 and acorner at x = −1.
Clint Lee Math 112 Lecture 1: Introduction to Limits 16/22
Defining Limits One-Sided Limits Infinite Limits
Example 4 – Calculating Limits
Continuing Example 4
Calculate each limit, if it exists. If the limit does not exist, explainwhy.
(a) (b)
(c) (d)
(e) (f)
Clint Lee Math 112 Lecture 1: Introduction to Limits 17/22
Defining Limits One-Sided Limits Infinite Limits
Example 4 – Calculating Limits
Continuing Example 4
Calculate each limit, if it exists. If the limit does not exist, explainwhy.
(a) limx→−1−
h(x) (b) limx→−1+
h(x)
(c) limx→−1
h(x) (d) limx→1−
h(x)
(e) limx→1+
h(x) (f) limx→1
h(x)
Clint Lee Math 112 Lecture 1: Introduction to Limits 17/22
Defining Limits One-Sided Limits Infinite Limits
Example 4 – Calculating Limits
Continuing Example 4
Calculate each limit, if it exists. If the limit does not exist, explainwhy.
(a) limx→−1−
h(x) = 6 (b) limx→−1+
h(x)
(c) limx→−1
h(x) (d) limx→1−
h(x)
(e) limx→1+
h(x) (f) limx→1
h(x)
Clint Lee Math 112 Lecture 1: Introduction to Limits 17/22
Defining Limits One-Sided Limits Infinite Limits
Example 4 – Calculating Limits
Continuing Example 4
Calculate each limit, if it exists. If the limit does not exist, explainwhy.
(a) limx→−1−
h(x) = 6 (b) limx→−1+
h(x) = 6
(c) limx→−1
h(x) (d) limx→1−
h(x)
(e) limx→1+
h(x) (f) limx→1
h(x)
Clint Lee Math 112 Lecture 1: Introduction to Limits 17/22
Defining Limits One-Sided Limits Infinite Limits
Example 4 – Calculating Limits
Continuing Example 4
Calculate each limit, if it exists. If the limit does not exist, explainwhy.
(a) limx→−1−
h(x) = 6 (b) limx→−1+
h(x) = 6
(c) limx→−1
h(x) = 6 (d) limx→1−
h(x)
(e) limx→1+
h(x) (f) limx→1
h(x)
Clint Lee Math 112 Lecture 1: Introduction to Limits 17/22
Defining Limits One-Sided Limits Infinite Limits
Example 4 – Calculating Limits
Continuing Example 4
Calculate each limit, if it exists. If the limit does not exist, explainwhy.
(a) limx→−1−
h(x) = 6 (b) limx→−1+
h(x) = 6
(c) limx→−1
h(x) = 6 (d) limx→1−
h(x) = 2
(e) limx→1+
h(x) (f) limx→1
h(x)
Clint Lee Math 112 Lecture 1: Introduction to Limits 17/22
Defining Limits One-Sided Limits Infinite Limits
Example 4 – Calculating Limits
Continuing Example 4
Calculate each limit, if it exists. If the limit does not exist, explainwhy.
(a) limx→−1−
h(x) = 6 (b) limx→−1+
h(x) = 6
(c) limx→−1
h(x) = 6 (d) limx→1−
h(x) = 2
(e) limx→1+
h(x) = 1 (f) limx→1
h(x)
Clint Lee Math 112 Lecture 1: Introduction to Limits 17/22
Defining Limits One-Sided Limits Infinite Limits
Example 4 – Calculating Limits
Continuing Example 4
Calculate each limit, if it exists. If the limit does not exist, explainwhy.
(a) limx→−1−
h(x) = 6 (b) limx→−1+
h(x) = 6
(c) limx→−1
h(x) = 6 (d) limx→1−
h(x) = 2
(e) limx→1+
h(x) = 1 (f) limx→1
h(x) DNE
Clint Lee Math 112 Lecture 1: Introduction to Limits 17/22
Defining Limits One-Sided Limits Infinite Limits
Example 4 – Calculating Limits
Continuing Example 4
Calculate each limit, if it exists. If the limit does not exist, explainwhy.
(a) limx→−1−
h(x) = 6 (b) limx→−1+
h(x) = 6
(c) limx→−1
h(x) = 6 (d) limx→1−
h(x) = 2
(e) limx→1+
h(x) = 1 (f) limx→1
h(x) DNE
The two-sided limit at x = 1 does not exist (DNE) because the twoone-sided limits at x = 1 are different.
Clint Lee Math 112 Lecture 1: Introduction to Limits 17/22
Defining Limits One-Sided Limits Infinite Limits
Infinite Limits
Consider the function
f (x) =x + 1
x2 − 1
from Example 1. This function takes arbitrarily large, positive andnegative, values near x = 1. So the function is undefined there. Wecan use limits to say this more precisely.
Clint Lee Math 112 Lecture 1: Introduction to Limits 18/22
Defining Limits One-Sided Limits Infinite Limits
Definition of Infinite Limits
Definition of Infinite Limits
Definition (Infinite Limits)
Definition of Two Sided Infinite Limits
Clint Lee Math 112 Lecture 1: Introduction to Limits 19/22
Defining Limits One-Sided Limits Infinite Limits
Definition of Infinite Limits
Definition of Infinite Limits
Definition (Infinite Limits)
We say thatlimx→a
f (x) = ∞
if f (x) can be made arbitrarily large and positive by making xsufficiently close to a, on either side of a,
Definition of Two Sided Infinite Limits
Clint Lee Math 112 Lecture 1: Introduction to Limits 19/22
Defining Limits One-Sided Limits Infinite Limits
Definition of Infinite Limits
Definition of Infinite Limits
Definition (Infinite Limits)
We say thatlimx→a
f (x) = ∞
if f (x) can be made arbitrarily large and positive by making xsufficiently close to a, on either side of a, and
limx→a
f (x) = −∞
if f (x) can be made arbitrarily large and negative by making xsufficiently close to a, on either side of a.
Definition of Two Sided Infinite Limits
Clint Lee Math 112 Lecture 1: Introduction to Limits 19/22
Defining Limits One-Sided Limits Infinite Limits
Definition of Infinite Limits
Definition of Infinite Limits
Definition (Infinite Limits)
We say thatlimx→a
f (x) = ∞
if f (x) can be made arbitrarily large and positive by making xsufficiently close to a, on either side of a, and
limx→a
f (x) = −∞
if f (x) can be made arbitrarily large and negative by making xsufficiently close to a, on either side of a.
Similar definitions apply for one-sided infinite limits.
Definition of Two Sided Infinite Limits
Clint Lee Math 112 Lecture 1: Introduction to Limits 19/22
Defining Limits One-Sided Limits Infinite Limits
Example 5 – Evaluating Infinite Limits
Example 5 – Evaluating Infinite Limits
For the function
f (x) =x + 1
x2 − 1
from Example 1 give the value as +∞ or −∞ for each limit, or explainwhy the infinite limit does not exist.
(a)
(b)
(c)
Clint Lee Math 112 Lecture 1: Introduction to Limits 20/22
Defining Limits One-Sided Limits Infinite Limits
Example 5 – Evaluating Infinite Limits
Example 5 – Evaluating Infinite Limits
For the function
f (x) =x + 1
x2 − 1
from Example 1 give the value as +∞ or −∞ for each limit, or explainwhy the infinite limit does not exist.
(a) limx→1−
f (x)
(b) limx→1+
f (x)
(c) limx→1
f (x)
Clint Lee Math 112 Lecture 1: Introduction to Limits 20/22
Defining Limits One-Sided Limits Infinite Limits
Example 5 – Evaluating Infinite Limits
Example 5 – Evaluating Infinite Limits
For the function
f (x) =x + 1
x2 − 1
from Example 1 give the value as +∞ or −∞ for each limit, or explainwhy the infinite limit does not exist.
(a) limx→1−
f (x) = −∞
(b) limx→1+
f (x)
(c) limx→1
f (x)
Clint Lee Math 112 Lecture 1: Introduction to Limits 20/22
Defining Limits One-Sided Limits Infinite Limits
Example 5 – Evaluating Infinite Limits
Example 5 – Evaluating Infinite Limits
For the function
f (x) =x + 1
x2 − 1
from Example 1 give the value as +∞ or −∞ for each limit, or explainwhy the infinite limit does not exist.
(a) limx→1−
f (x) = −∞
(b) limx→1+
f (x) = ∞
(c) limx→1
f (x)
Clint Lee Math 112 Lecture 1: Introduction to Limits 20/22
Defining Limits One-Sided Limits Infinite Limits
Example 5 – Evaluating Infinite Limits
Example 5 – Evaluating Infinite Limits
For the function
f (x) =x + 1
x2 − 1
from Example 1 give the value as +∞ or −∞ for each limit, or explainwhy the infinite limit does not exist.
(a) limx→1−
f (x) = −∞
(b) limx→1+
f (x) = ∞
(c) limx→1
f (x) DNE
Clint Lee Math 112 Lecture 1: Introduction to Limits 20/22
Defining Limits One-Sided Limits Infinite Limits
Example 5 – Evaluating Infinite Limits
Example 5 – Evaluating Infinite Limits
For the function
f (x) =x + 1
x2 − 1
from Example 1 give the value as +∞ or −∞ for each limit, or explainwhy the infinite limit does not exist.
(a) limx→1−
f (x) = −∞
(b) limx→1+
f (x) = ∞
(c) limx→1
f (x) DNE
The two-sided limit at x = 1 does not exist (DNE) because the twoone-sided limits at x = 1 are different.
Clint Lee Math 112 Lecture 1: Introduction to Limits 20/22
Appendix
More on Applying the Definition of Limit
In Example 2, if we wish to keep the value of f (x) within 0.03 of thelimit value, so that
−0.53 ≤ f (x) ≤ −0.47,
making the vertical extent of the box smaller, how much do we haveto decrease its horizontal extent?
Return
Clint Lee Math 112 Lecture 1: Introduction to Limits 21/22
Appendix
More on Applying the Definition of Limit
In Example 2, if we wish to keep the value of f (x) within 0.03 of thelimit value, so that
−0.53 ≤ f (x) ≤ −0.47,
making the vertical extent of the box smaller, how much do we haveto decrease its horizontal extent?Using the values from the table in Example 1 we see that we onlyneed to have x within 0.1 of the limit point, i.e., −1.1 ≤ x ≤ −0.9.
Return
Clint Lee Math 112 Lecture 1: Introduction to Limits 21/22
Appendix
More on Applying the Definition of Limit
In Example 2, if we wish to keep the value of f (x) within 0.03 of thelimit value, so that
−0.53 ≤ f (x) ≤ −0.47,
making the vertical extent of the box smaller, how much do we haveto decrease its horizontal extent?Using the values from the table in Example 1 we see that we onlyneed to have x within 0.1 of the limit point, i.e., −1.1 ≤ x ≤ −0.9.Since
f (−1.1) = −0.476190 < −0.47 and f (−0.9) = −0.526316 > −0.53
so that for x in the interval [−1.1,−0.9], the value of f is in theinterval [−0.526316,−0.476190], which is contained in the interval[−0.53,−0.47].
Return
Clint Lee Math 112 Lecture 1: Introduction to Limits 21/22
Appendix
More on Applying the Definition of Limit
In Example 2, if we wish to keep the value of f (x) within 0.03 of thelimit value, so that
−0.53 ≤ f (x) ≤ −0.47,
making the vertical extent of the box smaller, how much do we haveto decrease its horizontal extent?Using the values from the table in Example 1 we see that we onlyneed to have x within 0.1 of the limit point, i.e., −1.1 ≤ x ≤ −0.9.Since
f (−1.1) = −0.476190 < −0.47 and f (−0.9) = −0.526316 > −0.53
so that for x in the interval [−1.1,−0.9], the value of f is in theinterval [−0.526316,−0.476190], which is contained in the interval[−0.53,−0.47].You draw the graph, by hand and on your calculator or using Maple,similar to the one on the previous slide.
Return
Clint Lee Math 112 Lecture 1: Introduction to Limits 21/22
Appendix
Definition of One-Sided Infinite Limits
Definition (One-Sided Infinite Limits)
Return
Clint Lee Math 112 Lecture 1: Introduction to Limits 22/22
Appendix
Definition of One-Sided Infinite Limits
Definition (One-Sided Infinite Limits)
We say thatlim
x→a+f (x) = ∞
if f (x) can be made arbitrarily large and positive by making x suffi-ciently close to a with x > a,
Return
Clint Lee Math 112 Lecture 1: Introduction to Limits 22/22
Appendix
Definition of One-Sided Infinite Limits
Definition (One-Sided Infinite Limits)
We say thatlim
x→a+f (x) = ∞
if f (x) can be made arbitrarily large and positive by making x suffi-ciently close to a with x > a, and
limx→a−
f (x) = ∞
if f (x) can be made arbitrarily large and positive by making x suffi-ciently close to a with x < a.
Return
Clint Lee Math 112 Lecture 1: Introduction to Limits 22/22
Appendix
Definition of One-Sided Infinite Limits
Definition (One-Sided Infinite Limits)
We say thatlim
x→a+f (x) = −∞
if f (x) can be made arbitrarily large and negative by making x suffi-ciently close to a with x > a,
Return
Clint Lee Math 112 Lecture 1: Introduction to Limits 22/22
Appendix
Definition of One-Sided Infinite Limits
Definition (One-Sided Infinite Limits)
We say thatlim
x→a+f (x) = −∞
if f (x) can be made arbitrarily large and negative by making x suffi-ciently close to a with x > a, and
limx→a−
f (x) = −∞
if f (x) can be made arbitrarily large and negative by making x suffi-ciently close to a with x < a.
Return
Clint Lee Math 112 Lecture 1: Introduction to Limits 22/22
Appendix
Definition of One-Sided Infinite Limits
Definition (One-Sided Infinite Limits)
We say thatlim
x→a+f (x) = −∞
if f (x) can be made arbitrarily large and negative by making x suffi-ciently close to a with x > a, and
limx→a−
f (x) = −∞
if f (x) can be made arbitrarily large and negative by making x suffi-ciently close to a with x < a.
If the two one-sided infinite limits are different, then the two-sidedinfinite limit does not exist.
Return
Clint Lee Math 112 Lecture 1: Introduction to Limits 22/22