In the previous section we
looked at a couple of problems and in both problems we had a function (slope in
the tangent problem case and average rate of change in the rate of change
problem) and we wanted to know how that function was behaving at some point . At this stage of the game we no longer care
where the functions came from and we no longer care if we’re going to see them
down the road again or not. All that we
need to know or worry about is that we’ve got these functions and we want to
know something about them.
To answer the questions in the last section we choose values
of x that got closer and closer to and we plugged these into the function. We also made sure that we looked at values of
x that were on both the left and the
right of . Once we did this we looked at our table of
function values and saw what the function values were approaching as x got closer and closer to and used this to guess the value that we were
after.
This process is called taking
a limit and we have some notation for this.
The limit notation for the two problems from the last section is,
In this notation we will note that we always give the
function that we’re working with and we also give the value of x (or t) that we are moving in towards.
In this section we are going to take an intuitive approach
to limits and try to get a feel for what they are and what they can tell us
about a function. With that goal in mind
we are not going to get into how we actually compute limits yet. We will instead rely on what we did in the
previous section as well as another approach to guess the value of the limits.
Both of the approaches that we are going to use in this section
are designed to help us understand just what limits are. In general we don’t typically use the methods
in this section to compute limits and in many cases can be very difficult to
use to even estimate the value of a limit and/or will give the wrong value on
occasion. We will look at actually
computing limits in a couple of sections.
Let’s first start off with the following “definition” of a
limit.
Definition
We say that the limit of f(x) is L as x approaches a and write this as
provided we can make f(x)
as close to L as we want for all x sufficiently close to a, from both sides, without actually
letting x be a.

This is not the exact, precise definition of a limit. If you would like to see the more precise and
mathematical definition of a limit you should check out the The Definition of a Limit section at the end of this
chapter. The definition given above is
more of a “working” definition. This
definition helps us to get an idea of just what limits are and what they can
tell us about functions.
So just what does this definition mean? Well let’s suppose that we know that the
limit does in fact exist. According to
our “working” definition we can then decide how close to L that we’d like to make f(x). For sake of argument let’s suppose that we
want to make f(x) no more than 0.001
away from L. This means that we want one of the following
Now according to the “working” definition this means that if
we get x sufficiently close to a we can
make one of the above true. However, it
actually says a little more. It actually
says that somewhere out there in the world is a value of x, say X, so that for all
x’s that are closer to a than X then one of the above statements will be true.
This is actually a fairly important idea. There are many functions out there in the
world that we can make as close to L
for specific values of x that are
close to a, but there will be other
values of x closer to a that give functions values that are
nowhere near close to L. In order for a limit to exist once we get f(x) as close to L as we want for some x
then it will need to stay in that close to L
(or get closer) for all values of x
that are closer to a. We’ll see an example
of this later in this section.
In somewhat simpler terms the definition says that as x gets closer and closer to x=a (from both sides of course…) then f(x) must be getting closer and closer to L. Or, as we move in towards
x=a then f(x) must be moving in
towards L.
It is important to note once again that we must look at
values of x that are on both sides of
x=a.
We should also note that we are not allowed to use x=a in the definition. We
will often use the information that limits give us to get some information
about what is going on right at x=a,
but the limit itself is not concerned with what is actually going on at x=a.
The limit is only concerned with what is going on around the point x=a.
This is an important concept about limits that we need to keep in mind.
An alternative notation that we will occasionally use in
denoting limits is
How do we use this definition to help us estimate
limits? We do exactly what we did in the
previous section. We take x’s
on both sides of x=a that move in
closer and closer to a and we plug
these into our function. We then look to
see if we can determine what number the function values are moving in towards
and use this as our estimate.
Let’s work an example.
Example 1 Estimate
the value of the following limit.
Solution
Notice that I did say estimate the value of the
limit. Again, we are not going to
directly compute limits in this section.
The point of this section is to give us a better idea of how limits
work and what they can tell us about the function.
So, with that in mind we are going to work this in pretty
much the same way that we did in the last section. We will choose values of x that get closer and closer to x=2 and plug these values into the
function. Doing this gives the
following table of values.
x

f(x)

x

f(x)

2.5

3.4

1.5

5.0

2.1

3.857142857

1.9

4.157894737

2.01

3.985074627

1.99

4.015075377

2.001

3.998500750

1.999

4.001500750

2.0001

3.999850007

1.9999

4.000150008

2.00001

3.999985000

1.99999

4.000015000

Note that we made sure and picked
values of x that were on both sides
of and that we moved in very close to to make sure that any trends that we might
be seeing are in fact correct.
Also notice that we can’t actually
plug in into the function as this would give us a
division by zero error. This is not a
problem since the limit doesn’t care what is happening at the point in
question.
From this table it appears that the
function is going to 4 as x
approaches 2, so
.

Let’s think a little bit more about
what’s going on here. Let’s graph the
function from the last example. The
graph of the function in the range of x’s
that were interested in is shown below.
First, notice that there is a rather
large open dot at . This is there to remind us that the function
(and hence the graph) doesn’t exist at .
As we were plugging in values of x into the function we are in effect
moving along the graph in towards the point as . This is shown in the graph by the two arrows
on the graph that are moving in towards the point.
When we are computing limits the
question that we are really asking is what y
value is our graph approaching as we move in towards on our graph.
We are NOT asking what y value the graph takes at the point in
question. In other words, we are asking
what the graph is doing around the
point . In our case we can see that as x moves in towards 2 (from both sides)
the function is approaching even though the function itself doesn’t even
exist at . Therefore we can say that the limit is in
fact 4.
So what have we learned about limits? Limits are asking what the function is doing around and are not
concerned with what the function is actually doing at . This is a good thing as many of the functions
that we’ll be looking at won’t even exist at as we saw in our last example.
Let’s work another example to drive this point home.
Example 2 Estimate
the value of the following limit.
Solution
The first thing to note here is that this is exactly the
same function as the first example with the exception that we’ve now given it
a value for . So, let’s first note that
As far as estimating the value of this limit goes, nothing
has changed in comparison to the first example. We could build up a table of values as we
did in the first example or we could take a quick look at the graph of the
function. Either method will give us
the value of the limit.
Let’s first take a look at a table of values and see what
that tells us. Notice that the
presence of the value for the function at will not change our choices for x.
We only choose values of x
that are getting closer to but we never take . In other words the table of values that we
used in the first example will be exactly the same table that we’ll use
here. So, since we’ve already got it
down once there is no reason to redo it here.
From this table it is again clear that the limit is,
The limit is NOT
6! Remember from the discussion after
the first example that limits do not care what the function is actually doing
at the point in question. Limits are
only concerned with what is going on around
the point. Since the only thing about
the function that we actually changed was its behavior at this will not change the limit.
Let’s also take a quick look at this function's graph to
see if this says the same thing.
Again, we can see that as we move in towards on our graph the function is still
approaching a y value of 4. Remember that we are only asking what the
function is doing around and we don’t care what the function is
actually doing at . The graph then also supports the conclusion
that the limit is,

Let’s make the point one more time just to make sure we’ve
got it. Limits are not concerned with what is going on at .
Limits are only concerned with what is going on around . We keep saying this, but it is a very
important concept about limits that we must always keep in mind. So, we will take every opportunity to remind
ourselves of this idea.
Since limits aren’t concerned with what is actually
happening at we will, on occasion, see situations like the
previous example where the limit at a point and the function value at a point
are different. This won’t always happen
of course. There are times where the
function value and the limit at a point are the same and we will eventually see
some examples of those. It is important
however, to not get excited about things when the function and the limit do not
take the same value at a point. It
happens sometimes and so we will need to be able to deal with those cases when
they arise.
Let’s take a look another example to try and beat this idea
into the ground.
Example 3 Estimate
the value of the following limit.
Solution
First don’t get excited about the θ in function.
It’s just a letter, just like x
is a letter! It’s a Greek letter, but
it’s a letter and you will be asked to deal with Greek letters on occasion so
it’s a good idea to start getting used to them at this point.
Now, also notice that if we plug in θ =0 that we will get division by
zero and so the function doesn’t exist at this point. Actually, we get 0/0 at this point, but
because of the division by zero this function does not exist at θ =0.
So, as we did in the first example let’s get a table of
values and see what if we can guess what value the function is heading in
towards.




1

0.45969769

1

0.45969769

0.1

0.04995835

0.1

0.04995835

0.01

0.00499996

0.01

0.00499996

0.001

0.00049999

0.001

0.00049999

Okay, it looks like the function is moving in towards a
value of zero as θ moves in towards 0, from both sides of
course.
Therefore, the we will guess that the limit has the value,

So, once again, the limit had a value even though the
function didn’t exist at the point we were interested in.
It’s now time to work a couple of more examples that will
lead us into the next idea about limits that we’re going to want to discuss.
Example 4 Estimate
the value of the following limit.
Solution
Let’s build up a table of values and see what’s going on
with our function in this case.
t

f(t)

t

f(t)

1

1

1

1

0.1

1

0.1

1

0.01

1

0.01

1

0.001

1

0.001

1

Now, if we were to guess the limit from this table we
would guess that the limit is 1.
However, if we did make this guess we would be wrong. Consider any of the following function
evaluations.
In all three of these function evaluations we evaluated
the function at a number that is less than 0.001 and got three totally
different numbers. Recall that the
definition of the limit that we’re working with requires that the function be
approaching a single value (our guess) as t
gets closer and closer to the point in question. It doesn’t say that only some of the
function values must be getting closer to the guess. It says that all the function values must
be getting closer and closer to our guess.
To see what’s happening here a graph of the function would
be convenient.
From this graph we can see that as we move in towards the function starts oscillating wildly and
in fact the oscillations increases in speed the closer to that we get.
Recall from our definition of the limit that in order for a limit to
exist the function must be settling down in towards a single value as we get
closer to the point in question.
This function clearly does not settle in towards a single
number and so this limit does not
exist!

This last example points out the drawback of just picking
values of x using a table of function
values to estimate the value of a limit.
The values of x that we chose
in the previous example were valid and in fact were probably values that many
would have picked. In fact they were
exactly the same values we used in the problem before this one and they worked
in that problem!
When using a table of values there will always be the
possibility that we aren’t choosing the correct values and that we will guess
incorrectly for our limit. This is
something that we should always keep in mind when doing this to guess the value
of limits. In fact, this is such a problem
that after this section we will never use a table of values to guess the value
of a limit again.
This last example also has shown us that limits do not have
to exist. To this point we’ve only seen
limits that have existed, but that just doesn’t always have to be the case.
Let’s take a look at one more example in this section.
Example 5 Estimate
the value of the following limit.
Solution
This function is often called either the Heaviside or step function. We could
use a table of values to estimate the limit, but it’s probably just as quick
in this case to use the graph so let’s do that. Below is the graph of this function.
We can see from the graph that if we approach from the right side the function is moving
in towards a y value of 1. Well actually it’s just staying at 1, but
in the terminology that we’ve been using in this section it’s moving in
towards 1…
Also, if we move in towards from the left the function is moving in
towards a y value of 0.
According to our definition of the limit the function
needs to move in towards a single value as we move in towards (from both sides). This isn’t happening in this case and so in
this example we will also say that the limit doesn’t exist.
Note that the limit in this example is a little different
from the previous example. In the
previous example the function did not settle down to a single number as we
moved in towards . In this example however, the function does
settle down to a single number as on either side. The problem is that the number is different
on each side of . This is an idea that we’ll look at in a
little more detail in the next section.

Let’s summarize what we (hopefully) learned in this
section. In the first three examples we
saw that limits do not care what the function is actually doing at the point in
question. They only are concerned with
what is happening around the point. In
fact, we can have limits at even if the function itself does not exist at
that point. Likewise, even if a function
exists at a point there is no reason (at this point) to think that the limit
will have the same value as the function at that point. Sometimes the limit and the function will
have the same value at a point and other times they won’t have the same value.
Next, in the third and fourth examples we saw the main
reason for not using a table of values to guess the value of a limit. In those examples we used exactly the same
set of values, however they only worked in one of the examples. Using tables of values to guess the value of
limits is simply not a good way to get the value of a limit. This is the only section in which we will do
this. Tables of values should always be
your last choice in finding values of limits.
The last two examples showed us that not all limits will in
fact exist. We should not get locked
into the idea that limits will always exist.
In most calculus courses we work with limits that almost always exist
and so it’s easy to start thinking that limits always exist. Limits don’t always exist and so don’t get
into the habit of assuming that they will.
Finally, we saw in the fourth example that the only way to
deal with the limit was to graph the function.
Sometimes this is the only way, however this example also illustrated
the drawback of using graphs. In order
to use a graph to guess the value of the limit you need to be able to actually
sketch the graph. For many functions
this is not that easy to do.
There is another drawback in using graphs. Even if you actually have the graph it’s only
going to be useful if the y value is
approaching an integer. If the y value is approaching say there is no way that you’re going to be able
to guess that value from the graph and we are usually going to want exact
values for our limits.
So while graphs of functions can, on occasion, make your
life easier in guessing values of limits they are again probably not the best
way to get values of limits. They are
only going to be useful if you can get your hands on it and the value of the
limit is a “nice” number.
The natural question then is why did we even talk about
using tables and/or graphs to estimate limits if they aren’t the best way. There were a couple of reasons.
First, they can help us get a better understanding of what
limits are and what they can tell us. If
we don’t do at least a couple of limits in this way we might not get all that
good of an idea on just what limits are.
The second reason for doing limits in this way is to point
out their drawback so that we aren’t tempted to use them all the time!
We will eventually talk about how we really do limits. However, there is one more topic that we need
to discuss before doing that. Since this
section has already gone on for a while we will talk about this in the next
section.