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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 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 who 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.
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Example 2 Use
the definition of the limit to prove the following limit.
Solution
We’ll start this one out the same way that we did the
first one. We won’t be putting in
quite the same amount of explanation however.
Let’s start off by letting be any number then we need to find a number so that the following will be true.
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