In this section we will take a look at limits whose value is
infinity or minus infinity. These kinds
of limit will show up fairly regularly in later sections and in other courses
and so you’ll need to be able to deal with them when you run across them.
The first thing we should probably do here is to define just
what we mean when we say that a limit has a value of infinity or minus
infinity.
Definition
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We say
if we can make f(x)
arbitrarily large for all x
sufficiently close to x=a, from
both sides, without actually letting .
We say
if we can make f(x)
arbitrarily large and negative for all x
sufficiently close to x=a, from
both sides, without actually letting .
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These definitions can be appropriately modified for the
one-sided limits as well. To see a more
precise and mathematical definition of this kind of limit see the The Definition of the Limit section at the end of
this chapter.
Let’s start off with a fairly typical example illustrating
infinite limits.
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Example 1 Evaluate
each of the following limits.
Solution
So we’re going to be taking a look at a couple of
one-sided limits as well as the normal limit here. In all three cases notice that we can’t
just plug in . If we did we would get division by
zero. Also recall that the definitions
above can be easily modified to give similar definitions for the two
one-sided limits which we’ll be needing here.
Now, there are several ways we could proceed here to get
values for these limits. One way is to
plug in some points and see what value the function is approaching. In the proceeding section we said that we
were no longer going to do this, but in this case it is a good way to
illustrate just what’s going on with this function.
So, here is a table of values of x’s from both the left and the right. Using these values we’ll be able to
estimate the value of the two one-sided limits and once we have that done we
can use the fact that the normal
limit will exist only if the two one-sided limits exist and have the same
value.
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x
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x
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-0.1
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-10
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0.1
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10
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-0.01
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-100
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0.01
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100
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-0.001
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-1000
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0.001
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1000
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-0.0001
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-10000
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0.0001
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10000
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From this table we can see that as we make x smaller and smaller the function gets larger and larger and will retain the
same sign that x originally
had. It should make sense that this
trend will continue for any smaller value of x that we chose to use.
The function is a constant (one in this case) divided by an
increasingly small number. The
resulting fraction should be an increasingly large number and as noted above
the fraction will retain the same sign as x.
We can make the function as large and positive as we want
for all x’s sufficiently close to
zero while staying positive (i.e.
on the right). Likewise, we can make
the function as large and negative as we want for all x’s sufficiently close to zero while staying negative (i.e. on the left). So, from our definition above it looks like
we should have the following values for the two one sided limits.
Another way to see the values of the two one sided limits
here is to graph the function. Again,
in the previous section we mentioned that we won’t do this too often as most
functions are not something we can just quickly sketch out as well as the
problems with accuracy in reading values off the graph. In this case however, it’s not too hard to
sketch a graph of the function and, in this case as we’ll see accuracy is not
really going to be an issue. So, here
is a quick sketch of the graph.
So, we can see from this graph that the function does
behave much as we predicted that it would from our table values. The closer x gets to zero from the right the larger (in the positive sense)
the function gets, while the closer x
gets to zero from the left the larger (in the negative sense) the function
gets.
Finally, the normal limit, in this case, will not exist
since the two one-sided limits have different values.
So, in summary here are the values of the three limits for
this example.
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For most of the remaining examples in this section we’ll
attempt to “talk our way through” each limit.
This means that we’ll see if we can analyze what should happen to the
function as we get very close to the point in question without actually
plugging in any values into the function.
For most of the following examples this kind of analysis shouldn’t be
all that difficult to do. We’ll also
verify our analysis with a quick graph.
So, let’s do a couple more examples.
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Example 2 Evaluate
each of the following limits.
Solution
As with the previous example let’s start off by looking at
the two one-sided limits. Once we have
those we’ll be able to determine a value for the normal limit.
So, let’s take a look at the right-hand limit first and as
noted above let’s see if we can figure out what each limit will
be doing without actually plugging in any values of x into the function. As we
take smaller and smaller values of x,
while staying positive, squaring them will only make them smaller (recall
squaring a number between zero and one will make it smaller) and of course it
will stay positive. So we have a
positive constant divided by an increasingly small positive number. The result should then be an increasingly
large positive number. It looks like
we should have the following value for the right-hand limit in this case,
Now, let’s take
a look at the left hand limit. In this
case we’re going to take smaller and smaller values of x, while staying negative this time. When we square them they’ll get smaller, but
upon squaring the result is now positive.
So, we have a positive constant divided by an increasingly small
positive number. The result, as with
the right hand limit, will be an increasingly large positive number and so
the left-hand limit will be,
Now, in this
example, unlike the first one, the normal limit will exist and be infinity
since the two one-sided limits both exist and have the same value. So, in summary here are all the limits for
this example as well as a quick graph verifying the limits.
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With this next example we’ll move away from just an x in the denominator, but as we’ll see
in the next couple of examples they work pretty much the same way.
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Example 3 Evaluate
each of the following limits.
Solution
Let’s again start with the right-hand limit. With the right hand limit we know that we
have,
Also, as x gets closer and closer to -2 then will be getting closer and closer to
zero, while staying positive as noted above.
So, for the right-hand limit, we’ll have a negative constant divided
by an increasingly small positive number.
The result will be an increasingly large and negative number. So, it looks like the right-hand limit will
be negative infinity.
For the left
hand limit we have,
and will get closer and closer to zero (and be
negative) as x gets closer and
closer to -2. In this case then we’ll
have a negative constant divided by an increasingly small negative
number. The result will then be an
increasingly large positive number and so it looks like the left-hand limit
will be positive infinity.
Finally, since
two one sided limits are not the same the normal limit won’t exist.
Here are the
official answers for this example as well as a quick graph of the function
for verification purposes.
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At this point we should briefly acknowledge the idea of
vertical asymptotes. Each of the three
previous graphs have had one. Recall
from an Algebra class that a vertical asymptote is a vertical line (the dashed
line at in the previous example) in which the graph
will go towards infinity and/or minus infinity on one or both sides of the
line.
In an Algebra class they are a little difficult to define
other than to say pretty much what we just said. Now that we have infinite limits under our
belt we can easily define a vertical asymptote as follows,
Definition
Note that it only requires one of the above limits for a
function to have a vertical asymptote at .
Using this definition we can see that the first two examples
had vertical asymptotes at while the third example had a vertical
asymptote at .
We aren’t really going to do a lot with vertical asymptotes
here, but wanted to mention them at this point since we’d reached a good point to do
that.
Let’s now take a look at a couple more examples of infinite
limits that can cause some problems on occasion.
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Example 4 Evaluate
each of the following limits.
Solution
Let’s start with the right-hand limit. For this limit we have,
also, as . So, we have a positive constant divided by
an increasingly small negative number.
The results will be an increasingly large negative number and so it
looks like the right-hand limit will be negative infinity.
For the
left-handed limit we have,
and we still
have, as . In this case we have a positive constant
divided by an increasingly small positive number. The results will be an increasingly large
positive number and so it looks like the left-hand limit will be positive
infinity.
The normal
limit will not exist since the two one-sided limits are not the same. The official answers to this example are
then,
Here is a quick
sketch to verify our limits.
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All the examples to this point have had a constant in the
numerator and we should probably take a quick look at an example that doesn’t
have a constant in the numerator.
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Example 5 Evaluate
each of the following limits.
Solution
Let’s take a look at the right-handed limit first. For this limit we’ll have,
The main
difference here with this example is the behavior of the numerator as we let x get closer and closer to 3. In this case we have the following behavior
for both the numerator and denominator.
So, as we let x get closer and closer to 3 (always
staying on the right of course) the numerator, while not a constant, is
getting closer and closer to a positive constant while the denominator is
getting closer and closer to zero, and will be positive since we are on the
right side.
This means that
we’ll have a numerator that is getting closer and closer to a non-zero and
positive constant divided by an increasingly smaller positive number and so
the result should be an increasingly larger positive number. The right-hand limit should then be
positive infinity.
For the
left-hand limit we’ll have,
As with the right-hand limit we’ll have the following
behaviors for the numerator and the denominator,
The main difference in this case is that the denominator
will now be negative. So, we’ll have a
numerator that is approaching a positive, non-zero constant divided by an
increasingly small negative number.
The result will be an increasingly large and negative number.
The formal answers for this example are then,
As with most of
the examples in this section the normal limit does not exist since the two
one-sided limits are not the same.
Here’s a quick graph to verify our limits.
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So far all we’ve done is look at limits of rational
expressions, let’s do a couple of quick examples with some different functions.
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Example 6 Evaluate
Solution
First, notice that we can only evaluate the right-handed
limit here. We know that the domain of
any logarithm is only the positive numbers and so we can’t even talk about
the left-handed limit because that would necessitate the use of negative
numbers. Likewise, since we can’t deal
with the left-handed limit then we can’t talk about the normal limit.
This limit is pretty simple to get from a quick sketch of
the graph.
From this we can see that,
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Example 7 Evaluate
both of the following limits.
Solution
Here’s a quick sketch of the graph of the tangent
function.
From this it’s easy to see that we have the following
values for each of these limits,
Note that the
normal limit will not exist because the two one-sided limits are not the
same.
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We’ll leave this section with a few facts about infinite
limits.
Facts
To see the proof of this set of facts see the Proof of Various Limit
Properties section in the Extras chapter.
Note as well that the above set of facts also holds for
one-sided limits. They will also hold if
,
with a change of sign on the infinities in the first three parts. The proofs of these changes to the facts are
nearly identical to the proof of the original facts and so are left to the
reader.