In this section we’re going to be taking a look at the
precise, mathematical definition of the three kinds of limits we looked at in
this chapter. We’ll be looking at the
precise definition of limits at finite points that have finite values, limits
that are infinity and limits at infinity.
We’ll also give the precise, mathematical definition of continuity.
Let’s start this section out with the definition of a limit
at a finite point that has a finite value.
Wow. That’s a mouth
full. Now that it’s written down, just
what does this mean?
Let’s take a look at the following graph and let’s also
assume that the limit does exist.
What the definition is telling us is that for any number 
that we pick we can go to our graph and sketch
two horizontal lines at and as shown on the graph above. Then somewhere out there in the world is
another number ,
which we will need to determine, that will allow us to add in two vertical
lines to our graph at and .
Now, if we take any x
in the pink region, i.e. between and ,
then this x will be closer to a than either of and . Or,
If we now identify the point on the graph that our choice of
x gives then this point on the graph will lie in the intersection of the
pink and yellow region. This means that
this function value f(x) will be
closer to L than either of and . Or,
So, if we take any value of x in the pink region then the graph for those values of x will lie in the yellow region.
Notice that there are actually an infinite number of
possible δ ’s that we can choose. In fact, if we go back and look at the graph
above it looks like we could have taken a slightly larger δ and still gotten the graph from that pink
region to be completely contained in the yellow region.
Also, notice that as the definition points out we only need
to make sure that the function is defined in some interval around but we don’t really care if it is defined at . Remember that limits do not care what is
happening at the point, they only care what is happening around the point in
question.
Okay, now that we’ve gotten the definition out of the way
and made an attempt to understand it let’s see how it’s actually used in
practice.
These are a little tricky sometimes and it can take a lot of
practice to get good at these so don’t feel too bad if you don’t pick up on
this stuff right away. We’re going to be
looking at a couple of examples that work out fairly easily.
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Example 1 Use
the definition of the limit to prove the following limit.
Solution
In this case both L
and a are zero. So, let be any number. Don’t worry about what the number is, is just some arbitrary number. Now according to the definition of the
limit, if this limit is to be true we will need to find some other number so that the following will be true.
Or upon simplifying things we need,
Often the way to go through these is to start with the
left inequality and do a little simplification and see if that suggests a
choice for . We’ll start by bringing the exponent out of
the absolute value bars and then taking the square root of both sides.
Now, the results of this simplification looks an awful lot
like with the exception of the “ ” part. Missing that however isn’t a problem, it is
just telling us that we can’t take . So, it looks like if we choose we should get what we want.
We’ll next need to verify that our choice of will give us what we want, i.e.,
Verification is in fact pretty much the same work that we
did to get our guess. First, let’s
again let be any number and then choose . Now, assume that . We need to show that by choosing x to satisfy this we will get,
To start the verification process we’ll start with and then first strip out the exponent from
the absolute values. Once this is done
we’ll use our assumption on x,
namely that . Doing all this gives,
Or, upon taking
the middle terms out, if we assume that then we will get,
and this is exactly what we needed to show.
So, just what have we done? We’ve shown that if we choose then we can find a so that we have,
and according to our definition this means that,
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These can be a little tricky the first couple times
through. Especially when it seems like
we’ve got to do the work twice. In the
previous example we did some simplification on the left hand inequality to get
our guess for and then seemingly went through exactly the
same work to then prove that our guess was correct. This is often how these work, although we
will see an example here in a bit where things don’t work out quite so nicely.
So, having said that let’s take a look at a slightly more
complicated limit, although this one will still be fairly similar to the first
example.
Okay, so again the process seems to suggest that we have to
essentially redo all our work twice, once to make the guess for and then another time to prove our
guess. Let’s do an example that doesn’t
work out quite so nicely.
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Example 3 Use
the definition of the limit to prove the following limit.
Solution
So, let’s get started.
Let be any number then we need to find a number so that the following will be true.
We’ll start the
guess process in the same manner as the previous two examples.
Okay, we’ve
managed to show that is equivalent to . However, unlike the previous two examples,
we’ve got an extra term in here that doesn’t show up in the right inequality
above. If we have any hope of
proceeding here we’re going to need to find some way to get rid of the .
To do this
let’s just note that if, by some chance, we can show that for some number K then, we’ll have the following,
If we now
assume that what we really want to show is instead of we get the following,
This is
starting to seem familiar isn’t it?
All this work
however, is based on the assumption that we can show that for some K. Without this assumption we can’t do
anything so let’s see if we can do this.
Let’s first
remember that we are working on a limit here and let’s also remember that
limits are only really concerned with what is happening around the point in
question, in this case. So, it is safe to assume that whatever x is, it must be close to . This means we can safely assume that
whatever x is, it is within a
distance of, say one of . Or in terms of an inequality, we can assume
that,
Why choose 1
here? There is no reason other than
it’s a nice number to work with. We
could just have easily chosen 2, or 5, or . The only difference our choice will make is
on the actual value of K that we
end up with. You might want to go
through this process with another choice of K and see if you can do it.
So, let’s start
with and get rid of the absolute value bars and
this solve the resulting inequality for x
as follows,
If we now add 5
to all parts of this inequality we get,
Now, since (the positive part is important here) we can
say that, provided we know that . Or, if take the double inequality above we
have,
So, provided we can see that which in turn gives us,
So, to this
point we make two assumptions about We’ve assumed that,
It may not seem
like it, but we’re now ready to chose a . In the previous examples we had only a
single assumption and we used that to give us . In this case we’ve got two and they BOTH
need to be true. So, we’ll let be the smaller of the two assumptions, 1 and
. Mathematically, this is written as,
By doing this
we can guarantee that,
Now that we’ve
made our choice for we need to verify it. So, be any number and then choose . Assume that . First, we get that,
We also get,
Finally, all we
need to do is,
We’ve now
managed to show that,
and so by our
definition we have,
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Okay, that was a lot more work that the first two examples
and unfortunately, it wasn’t all that difficult of a problem. Well, maybe we should say that in comparison
to some of the other limits we could have tried to prove it wasn’t all that
difficult. When first faced with these
kinds of proofs using the precise definition of a limit they can all seem
pretty difficult.
Do not feel bad if you don’t get this stuff right away. It’s very common to not understand this right
away and to have to struggle a little to fully start to understand how these
kinds of limit definition proofs work.
Next, let’s give the precise definitions for the right- and
left-handed limits.
Note that with both of these definitions there are two ways
to deal with the restriction on x and
the one in parenthesis is probably the easier to use, although the main one
given more closely matches the definition of the normal limit above.
Let’s work a quick example of one of these, although as
you’ll see they work in much the same manner as the normal limit problems do.
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Example 4 Use
the definition of the limit to prove the following limit.
Solution
Let be any number then we need to find a number so that the following will be true.
Or upon a
little simplification we need to show,
As with the
previous problems let’s start with the left hand inequality and see if we
can’t use that to get a guess for . The only simplification that we really need
to do here is to square both sides.
So, it looks
like we can chose .
Let’s verify
this. Let be any number and chose . Next assume that . This gives,
We now shown
that,
and so by the definition of the right-hand limit we have,
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Let’s now move onto the definition of infinite limits. Here are the two definitions that we need to
cover both possibilities, limits that are positive infinity and limits that are
negative infinity.
In these two definitions note that M must be a positive number and that N must be a negative number.
That’s an easy distinction to miss if you aren’t paying close
attention.
Also note that we could also write down definitions for
one-sided limits that are infinity if we wanted to. We’ll leave that to you to do if you’d like
to.
Here is a quick sketch illustrating Definition 4.
What Definition 4 is telling us is that no matter how large
we choose M to be we can always find
an interval around ,
given by for some number ,
so that as long as we stay within that interval the graph of the function will
be above the line as shown in the graph. Also note that we don’t need the function to
actually exist at in order for the definition to hold. This is also illustrated in the sketch above.
Note as well that the larger M is the smaller we’re probably going to need to make .
To see an illustration of Definition 5 reflect the above
graph about the x-axis and you’ll see
a sketch of Definition 5.
Let’s work a quick example of one of these to see how these
differ from the previous examples.
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Example 5 Use
the definition of the limit to prove the following limit.
Solution
These work in pretty much the same manner as the previous
set of examples do. The main
difference is that we’re working with an M
now instead of an . So, let’s get going.
Let be any number and we’ll need to choose a so that,
As with the all
the previous problems we’ll start with the left inequality and try to get
something in the end that looks like the right inequality. To do this we’ll basically solve the left
inequality for x and we’ll need to
recall that . So, here’s that work.
So, it looks
like we can chose . All we need to do now is verify this guess.
Let be any number, choose and assume that .
In the previous
examples we tried to show that our assumptions satisfied the left inequality
by working with it directly. However,
in this, the function and our assumption on x that we’ve got actually will make this easier to start with the
assumption on x and show that we
can get the left inequality out of that.
Note that this is being done this way mostly because of the function
that we’re working with and not because of the type of limit that we’ve got.
Doing this work
gives,
So, we’ve
managed to show that,
and so by the
definition of the limit we have,
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For our next set of limit definitions let’s take a look at
the two definitions for limits at infinity.
Again, we need one for a limit at plus infinity and another for negative
infinity.
To see what these definitions are telling us here is a quick
sketch illustrating Definition 6.
Definition 6 tells us is that no matter how close to L we want to get, mathematically this is
given by for any chosen ,
we can find another number M such
that provided we take any x bigger
than M, then the graph of the
function for that x will be closer to
L than either and . Or, in other words, the graph will be in the
shaded region as shown in the sketch below.
Finally, note that the smaller we make the larger we’ll probably need to make M.
Here’s a quick example of one of these limits.
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Example 6 Use
the definition of the limit to prove the following limit.
Solution
Let be any number and we’ll need to choose a so that,
Getting our
guess for N isn’t too bad here.
Since we’re
heading out towards negative infinity it looks like we can choose . Note that we need the “-” to make sure
that N is negative (recall that ).
Let’s verify
that our guess will work. Let and choose and assume that . As with the previous example the function
that we’re working with here suggests that it will be easier to start with
this assumption and show that we can get the left inequality out of that.
Note that when
we took the absolute value of both sides we changed both sides from negative
numbers to positive numbers and so also had to change the direction of the
inequality.
So, we’ve shown
that,
and so by the
definition of the limit we have,
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For our final limit definition let’s look at limits at
infinity that are also infinite in value.
There are four possible limits to define here. We’ll do one of them and leave the other
three to you to write down if you’d like to.
The other three definitions are almost identical. The only differences are the signs of M and/or N and the corresponding inequality directions.
As a final definition in this section let’s recall that we
previously said that a function was continuous if,
So, since continuity, as we previously defined it, is
defined in terms of a limit we can also now give a more precise definition of
continuity. Here it is,
This definition is very similar to the first definition in
this section and of course that should make some sense since that is exactly
the kind of limit that we’re doing to show that a function is continuous. The only real difference is that here we need
to make sure that the function is actually defined at ,
while we didn’t need to worry about that for the first definition since limits
don’t really care what is happening at the point.
We won’t do any examples of proving a function is continuous
at a point here mostly because we’ve already done some examples. Go back and look at the first three
examples. In each of these examples the
value of the limit was the value of the function evaluated at and so in each of these examples not only did
we prove the value of the limit we also managed to prove that each of these
functions are continuous at the point in question.