3.1: Limits(2 of 2)
Lecture 6
MTH 124
Learning Objectives
Procedural Skills
1. Recognize limit notation and understand a question involving a limit is asking.
2. Solve a variety of limit problems using graphical and numerical techniques.
Interpretation Skills
1. Give examples of functions where we cannot evaluate the function at the limit but where
the limit does exist.
2. Synthesize the relationship between limit, AROC, and instantaneous rate of change.
Warm-Up
1. (a) Determine lim 5h + 2.
h→0
(b) Sketch a function R(x) that satisfies the condition lim R(x)=20.
x→3
(c) Sketch a function P (t) whose limit is 4 as t approaches infinity.
33
3.1: Limits(2 of 2)
Lecture 6
MTH 124
(d) Sketch a function g(t) that satisfies the following conditions.
• g(t) approaches infinity as t approaches negative infinity, and
• g(t) approaches 0 as t approaches infinity.
(e) Sketch or give an example of a function f (x) where lim f (x) = −∞ but f (0) is not
x→0
defined.
(f) Sketch a function k(x) where k(5) = 2 and lim k(x) = 10.
x→5
34
3.1: Limits(2 of 2)
Lecture 6
MTH 124
Limits at Infinity
A natural question related to limits is to ask what end behavior a function has. To understand
this in more detail let us consider the logistic equation as an example, which is given by
L
1 + Ae−kt
This equation is often used to model population growth or decay.1 Given the context of
population growth/decay, the variables above can be understood as the following:
y(t) =
y : The population of whatever we might be analyzing
L : The carrying capacity
A : A constant related to the rate of the growth or decay
k : A constant related to the rate of the growth or decay
t : Time
Now, suppose we had a population of Emperor Penguins in Antarctica, which we learn, after
careful study, can be modeled by the equation
y(t) =
400, 000
,
1 + 0.143e−0.01t
where t is years after the year 2010. We have previously used equations to answer questions
such as
“What is the initial population?”
“What will the population be in the year 2020?”
“When will the population reach 300,000 penguins?”
However, another meaningful question we could ask is what will happen to the penguin population
in the long term?
Put in context, based on our model, how many penguins can Antarctica support?
This sounds an awful lot like end behavior. Mathematically this is asking what is happening
to our population function as as time approaches infinity. Symbolically this is asking us to
evaluate
400, 000
.
t→∞ 1 + 0.143e−0.01t
First let’s notice that the term e−0.01t approaches 0 as t approaches infinity.2 Since e−0.01t
approaches zero, we can see the term 1 + 0.143e−0.01t approaches 1 so the limit above approaches
400,000 Emperor Penguins. From this limit we can see why the variable L is called carrying
capacity; it represents the maximum population that the model can sustain.
lim
1
This equation gives us an exponential growth/decay phase followed by a plateau, which is much more
physically realistic in most contexts.
2
Use a plot or table to see this.
35
3.1: Limits(2 of 2)
Lecture 6
MTH 124
The Limit of our Intuition3
To develop a strong intuition about limits it is important to consider many different examples.
In particular it may be helpful to remind yourself of the following ideas while you practice
solving problems involving limits.
• In general lim f (x) is not equal to the value of f (a)
x→a
• A limit can be equal to ±∞
• A limit may not exist
a
• Given a limit of the form lim f (x), what does is mean for x to approach ±∞?
x→±∞
a
For example consider lim g(x) where g(x) = x1 .
x→0
Taking the Limit of an Average Rate of Change
Recall our major motivation to develop an understanding of limits had its genesis in our
conversation about using AROC to determine instantaneous rate of change. We can now
approach this problem.
f (x + h) − f (x)
where f (x) = x2 + 3x at the point
Example 1 Determine the lim
h→0
h
x = 1.
Before we proceed with the procedure of solving this problem it is worth interpreting
f (x + h) − f (x)
what this question is asking us. Notice the term
is the average rate
h
of change of f (x) over the interval [x, x + h]. By taking x = 1 we have
lim
h→0
f (1 + h) − f (1)
f (x + h) − f (x)
= lim
.
h→0
h
h
So the expression above is is the limit of the average rate of change of f (x) over
[1, 1 + h] as h (our interval width) approaches zero. In other words, this is the
instantaneous rate of change of f (x) at x = 1. So the answer to this question
tells us exactly how f (x) is increasing/decreasing at the point x = 1.
To determine this limit we need to first determine the algebraic expression for
f (1) and f (1 + h). These are given by
f (1) =
, and
This gives us
f (1 + h) − f (1)
=
h→0
h
lim
3
Terrible pun intended!
36
f (1 + h) =
3.1: Limits(2 of 2)
Lecture 6
Thus we have that
MTH 124
f (1 + h) − f (1)
=
h→0
h
lim
Practice this type of problem by solving the following AROC limit on your own.
f (x + h) − f (x)
where f (x) = 3x − 8 at the point x = 4.
h→0
h
2. (a) Determine4 the lim
(b) Which of the following statements is true?
(a)
(b)
(b)
(c)
(d)
The limit above gives us the value of f (x) at the point x = 4.
The limit above gives us the AROC f (x) over the interval [4, 4 + h].
The limit above gives us an estimate of the AROC f (x) over the interval [4, 4 + h].
The limit above gives the instantaneous rate of change of f (x) at x = 4.
The limit above gives us an estimate of the instantaneous rate of change of f (x)
at x = 4
Counterintuitive Examples of Limits
Below are a few famous examples that highlight the sometimes counter-intuitive nature of limits.
Achilles and the Tortise
Zeno’s paradox5
Is 0.999.. equal to 1?
Impress Your Friends!6
4
Whenever solving problems of this type, simplify the quotient as much as possible before you attempt to
take the limit of the expression.
5
Hyperlink: https://www.youtube.com/watch?v=skM37PcZmWE
6
Hyperlink: https://www.youtube.com/watch?v=u7Z9UnWOJNY
37