More on the Limits of Functions
The basic concept upon which
calculus depends is the limit of a function. Consider a circular
plastic plate that is expanding uniformly because it is being heated. If
the radius is represented by the symbol, x, the area of the plate is given by 
. [A is the
function.]
As x gets closer and closer to 3
inches, the area, A, gets closer and closer to 
square
inches. Another way saying this is “as x approaches 3, the value of 
approaches as a limit.”
A symbolic
way of saying the same thing is 
.
If 
is a function and 
is a number, then
the notation 
means that 
gets closer and
closer to the real number 
as 
gets closer and
closer to the real number .
Example 1:
where 
is
the function and 35 is the limit of the function as x gets closer and closer to
5.
Example
2: 
where 
is
the function and 25 is the limit of the function as gets
closer and closer to 3.
In examples 1 and 2, we determined the limit of the function by simple arithmetic substitution. Many times we cannot determine the limit of a function by this type of simple arithmetic substitution. The function values may jump around so that they never settle down and approach a limit, in which case we say that the limit does not exist. The function may also be so complicated that the limit, even thought it exists, is not evident by this type of substitution.
Example 3: If we try to
calculate 
by the means we used
in examples 1 and 2, we get 
, which has no
meaning. We cannot say that a limit exists and we cannot say that a limit
does not exist.
So how do we find the limit of the function at 
when the
function is not defined at ? We start by
evaluating the function with numbers to the left of 2 and to the right of
2. We are interested in the behavior of the values of the function as the
x gets closer and closer to 2.
Let’s first examine the function
values to the left of as the values of get closer and
closer to 2.
|
|
1 |
1.5 |
1.8 |
1.9 |
1.99 |
1.999 |
1.9999 |
|
|
5 |
6.50 |
7.40 |
7.70 |
7.97 |
7.997 |
7.9997 |
According to the above table, it
appears that as x approaches the value of 2 from the left, the function value
is getting closer and closer to 8. A symbolic way of saying this is 
.
Let’s next examine the function
values to the right of as the values get
closer and closer to 2.
|
|
3 |
2.5 |
2.2 |
2.1 |
2.01 |
2.001 |
2.00001 |
|
|
11 |
9.50 |
8.60 |
8.30 |
8.03 |
8.003 |
8.00003 |
According to the above table, it appears
that as x approaches the value of 2 from the right, the function value is
getting closer and closer to 8. A symbolic way of saying this is 
.
Since and , we say
that 
.
Let’s rewrite the same problem as
.
Recall that 
as
long as
. What does this
graph look like? It looks like a line with a hole at 
.
In the above example, it did not matter that the function was not defined at 2. The only thing that did matter was how the function is defined (behaved) for values of x near 2.
Example 4: Note that there
is no problem in calculating 
because the function
is defined for 
. 
We are now going to talk about a
more formal definition of limit. To say, “ comes
closer and closer to the real number as
x comes closer and closer to the real number a” is not precise enough.
How close is to close enough?
How close is to 
close enough?
You might think “who cares?” As it turns out, many people do. Suppose, for example, you are the director of a fiber optics project. You have to guarantee to your client a specific flow rate of 10 seconds plus or minus 0.0001 seconds. The thickness of the cable is directly proportional to the expansion/contraction possibilities of the cable. How thick/thin should the cable be to guarantee that the flow will be within 0.001 seconds of the required rate (limit)? You could tell your client that if the radius of the cable is .25 millimeters plus or minus 0.0015 millimeters, you could guarantee that the flow will be within plus or minus 0.0001 seconds of the required 10 seconds.
Let’s talk about what how we
symbolize the values 0.0015 and 0.0001 so that they can be easily changed in a
discussion. The 0.0015 will represent “delta” and the symbol for “delta”
is 
. The 0.0001 will
represent “epsilon” and the symbol for “epsilon” is 
.
For this problem, we say that if the radius is in the interval
,
we can guarantee the flow rate (limit) is within the interval
.
In terms of delta and epsilon, we
write if 
, then 
.
Yet another way of saying the same
thing is that whenever 
, where is the radius, then 
, where is time.
This means that if the function has
a limit at a particular value, we can require that can be
as small as we want, and we will always find a that
will guarantee it.
Let’s now write a formal definition of the limit of a function.
Let be
a function defined on some open interval containing the number , except possibly at the
number itself. For example, the open interval 
contains
the number 3. However, we can also say we want the set of all the real
numbers in the interval except the number
3. The limit could still exist at 
even
thought the function is not defined at .
The statement 
means
that, for each positive number,, there exists a positive
number, , such that 
holds whenever 
.
Example 5: Let’s go back
to example 1, 
, and
require the 
. We are asking for
what interval 
can we guarantee
that the values of will be in the
interval 
.
Let 
.
This statement can represented by the absolute inequality 
which
can be restated as 
and finally as 
.
Let’s now see if we can work the
inequality so that the center
of the inequality is 
or 
.
Remember we want 
. Let
and solve for .
This tells us that if we choose any number in the interval
, the
function value will be within 0.00001 units of the limit of 35.
Let’s check this out at 
. 
.
Note that 
and
0.000005 is less than the required epsilon of 0.00001.
A Graphical Explanation:
Sketch and
sketch the lines 
and . Let’s find the
points of determine where the graph of intersects
the graphs of and .
In order to have the function
value fall within units of 35, we must
require that the values of fall within 
units of 5.
Therefore, 
If we require that , Example 5 above, then , same as we found above.
Example 6:
Let’s prove that the limit of the function, 
, as x approaches 4, is
22. Another way of saying this is let’s prove 
.
To prove , we have to show that for
any given real number,, no matter how small,
there is a real number,, such that when we
evaluate the function with a number within the interval 
,
the value of the function will always be in the interval 
.
A symbolic way of saying the same thing is 
whenever 
.
This means if we choose the , we can
guarantee that the limit of the function will be within units
of 22.
Let’s illustrate this with an example:
Suppose we require the limit of the function be within 0.0001 units of 22. How tight an interval around 4 do we need to achieve this?
Since , set 
.
If we evaluate the function at any real number in the interval 
, the function value will
be within 0.0001 units of 22.
and 
A graphical look at the problem is as follows:
Find the point where the graph of crosses
the graph of 
and 
.
This shows that the distance between the graph of the
function and each of the horizontal lines and
is 
whenever 
. This means that .
Example 7:
Prove 
.
Clearly 
does not
exist. So what we want to show is that the function values for numbers
close to 
are close to 8.
To prove this we have to show that for any given real
number,, no matter how small,
there is a real number,, such that
when we evaluate the function with a number within the interval 
, the value of the
function will be in the interval 
.
A graphical look at the problem is as follows:
Find the point where the graph of 
crosses
the graph of 
and 
.
This shows that the distance between the graph of the
function and each of the horizontal lines and
is when 
. This means that 
.
Let’s verify this with an example: How tight an
interval do we need around 
so that the value of
the function will be within 0.0001 units of 8. According to
the above, since , if we set up an interval
within 0.0001 of -4, the function value will be within 0.0001 units of 8.
Let’s choose 
because
and
What about functions with no limits?
Consider the function 
.
Graph this function and you will see a gap in the graph at .
This means that the limit of the function at does
not exist. How do we prove this?
We start out by assuming
that the limit does exist at and when we find a
contradiction, we know that our initial assumption is incorrect. If our
initial assumption that the function exists is not true, then the only other
alternative is that the limit of the function does not exit at .
Assume 
, where L is a real
number. This means we can choose any value for and
we know that there is a value such that 
, where is in the interval 
, will be within units of L.
For the sake of an example, let’s choose 
.
If the limit exists, there exist a such
that 
whenever 
.
- To the right of ,
. Therefore, 
.
This means that 
or
.
- To the
left of ,
and
. This means that 
or 
.
From the above we require that and . Since this is
impossible, our initial assumption that the limit exists at is
incorrect. There is no limit to this function as x approaches the value
of 2.