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In the previous section we looked at limits at infinity of
polynomials and/or rational expression involving polynomials. In this section we want to take a look at
some other types of functions that often show up in limits at infinity. The functions we’ll be looking at here are
exponentials, natural logarithms and inverse tangents.
Let’s start by taking a look at a some of very basic
examples involving exponential functions.
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Example 1 Evaluate
each of the following limits.
Solution
There are really just restatements of facts given in the basic exponential section of the review so we’ll
leave it to you to go back and verify these.
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The main point of this example was to point out that if the
exponent of an exponential goes to infinity in the limit then the exponential
function will also go to infinity in the limit.
Likewise, if the exponent goes to minus infinity in the limit then the
exponential will go to zero in the limit.
Here’s a quick set of examples to illustrate these ideas.
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Example 2 Evaluate
each of the following limits.
(a) [Solution]
(b) [Solution]
(c) [Solution]
Solution
(a)
In this part what we need to note (using Fact 2 above) is
that in the limit the exponent of the exponential does the following,
So, the
exponent goes to minus infinity in the limit and so the exponential must go
to zero in the limit using the ideas from the previous set of examples. So, the answer here is,
[Return to Problems]
(b)
Here let’s first note that,
The exponent
goes to infinity in the limit and so the exponential will also need to go to
infinity in the limit. Or,
[Return to Problems]
(c)
On the surface this part doesn’t appear to belong in this
section since it isn’t a limit at infinity.
However, it does fit into the ideas we’re examining in this set of
examples.
So, let’s first note that using the idea from the previous
section we have,
Remember that
in order to do this limit here we do need to do a right hand limit.
So, the
exponent goes to infinity in the limit and so the exponential must also go to
infinity.
Here’s the
answer to this part.
[Return to Problems]
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Let’s work some more complicated examples involving
exponentials. In the following set of
examples it won’t be that the exponents are more complicated, but instead that
there will be more than one exponential function to deal with.
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Example 3 Evaluate
each of the following limits.
(a) [Solution]
(b) [Solution]
Solution
So, the only difference between these two limits is the
fact that in the first we’re taking the limit as we go to plus infinity and
in the second we’re going to minus infinity.
To this point we’ve been able to “reuse” work from the first limit in
the at least a portion of the second limit.
With exponentials that will often not be the case we we’re going to
treat each of these as separate problems.
(a)
Let’s start by just taking the limit of each of the pieces
and see what we get.
The last two
terms aren’t any problem (they will be in the next part however, do you see
that?). The first three are a problem
however as they present us with another indeterminate form.
When dealing
with polynomials we factored out the term with the largest exponent in
it. Let’s do the same thing here. However, we now have to deal with both
positive and negative exponents and just what do we mean by the “largest”
exponent. When dealing with these here
we look at the terms that are causing the problems and ask which is the
largest exponent in those terms. So,
since only the first three terms are causing us problems (i.e. they all evaluate to an infinity
in the limit) we’ll look only at those.
So, since 10x is the largest of the three
exponents there we’ll “factor” an out of the whole thing. Just as with polynomials we do the
factoring by, in essence, dividing each term by and remembering that to simply the division
all we need to do is subtract the exponents.
For example, let’s just take a look at the last term,
Doing factoring
on all terms then gives,
Notice that in
doing this factoring all the remaining exponentials now have negative
exponents and we know that for this limit (i.e. going out to positive infinity) these will all be zero in
the limit and so will no longer cause problems.
We can now take
the limit and doing so gives,
To simplify the
work here a little all we really needed to do was factor the out of the “problem” terms (the first three
in this case) as follows,
We factored the
out of all terms for the practice of
doing the factoring and to avoid any issues with having the extra terms at
the end.
[Return to Problems]
(b)
Let’s start this one off in the same manner as the first
part. Let’s take the limit of each of
the pieces. This time note that
because our limit is going to negative infinity the first three exponentials
will in fact go to zero (because their exponents go to minus infinity in the
limit). The final two exponentials
will go to infinity in the limit (because their exponents go to plus infinity
in the limit).
Taking the limits gives,
So, the last
two terms are the problem here as they once again leave us with an
indeterminate form. As with the first
example we’re going to factor out the “largest” exponent in the last two
terms. This time however, “largest”
doesn’t refer to the bigger of the two numbers (-2 is bigger than -15). Instead we’re going to use “largest” to
refer to the exponent that is farther away from zero. Using this definition of “largest” means
that we’re going to factor an out.
Again, remember
that to factor this out all we really are doing is dividing each term by and then subtracting exponents. Here’s the work for the first term as an
example,
As with the
first part we can either factor it out of only the “problem” terms (i.e. the last two terms), or all the
terms. For the practice we’ll factor
it out of all the terms. Here is the
factoring work for this limit,
Finally, all we
need to do is take the limit.
[Return to Problems]
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So, when dealing with sums and/or differences of exponential
functions we look for the exponential with the “largest” exponent and remember
here that “largest” means the exponent farthest from zero. Also remember that if we’re looking at a
limit at plus infinity only the exponentials with positive exponents are going
to cause problems so those are the only terms we look at in determining the
largest exponent. Likewise, if we are
looking at a limit at minus infinity then only exponentials with negative
exponents are going to cause problems and so only those are looked at in
determining the largest exponent.
Finally, as you might have been able to guess from the
previous example when dealing with a sum and/or difference of exponentials all
we need to do is look at the largest exponent to determine the behavior of the
whole expression. Again, remembering
that if the limit is at plus infinity we only look at exponentials with
positive exponents and if we’re looking at a limit at minus infinity we only
look at exponentials with negative exponents.
Let’s next take a look at some rational functions involving
exponentials.
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Example 4 Evaluate
each of the following limits.
(a) [Solution]
(b) [Solution]
(c) [Solution]
Solution
As with the previous example, the only difference between
the first two parts is that one of the limits is going to plus infinity and
the other is going to minus infinity and just as with the previous example
each will need to be worked differently.
(a)
The basic concept involved in working this problem is the
same as with rational expressions in the previous section. We look at the denominator and determine
the exponential function with the “largest” exponent which we will then
factor out from both numerator and denominator. We will use the same reasoning as we did
with the previous example to determine the “largest” exponent. In the case since we are looking at a limit
at plus infinity we only look at exponentials with positive exponents.
So, we’ll factor an out of both then numerator and
denominator. Once that is done we can
cancel the and then take the limit of the remaining
terms. Here is the work for this
limit,
[Return to Problems]
(b)
In this case we’re going to minus infinity in the limit
and so we’ll look at exponentials in the denominator with negative exponents
in determining the “largest” exponent.
There’s only one however in this problem so that is what we’ll
use.
Again, remember to only look at the denominator. Do NOT use the exponential from the
numerator, even though that one is “larger” than the exponential in then
denominator. We always look only at
the denominator when determining what term to factor out regardless of what
is going on in the numerator.
Here is the work for this part.
[Return to
Problems]
(c)
We’ll do the work on this part with much less detail.
[Return to Problems]
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Next, let’s take a quick look at some basic limits involving
logarithms.
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Example 5 Evaluate
each of the following limits.
Solution
As with the last example I’ll leave it to you to verify
these restatements from the basic logarithm section.
Note that we had to do a right-handed limit for the first
one since we can’t plug negative x’s
into a logarithm. This means that the
normal limit won’t exist since we must look at x’s from both sides of the point in question and x’s to the left of zero are negative.
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From the previous example we can see that if the argument of
a log (the stuff we’re taking the log of) goes to zero from the right (i.e. always positive) then the log goes
to negative infinity in the limit while if the argument goes to infinity then
the log also goes to infinity in the limit.
Note as well that we can’t look at a limit of a logarithm as
x approaches minus infinity since we
can’t plug negative numbers into the logarithm.
Let’s take a quick look at some logarithm examples.
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Example 6 Evaluate
each of the following limits.
(a) [Solution]
(b) [Solution]
Solution
(a)
So, let’s first look to see what the argument of log is
doing,
The argument of
the log is going to infinity and so the log must also be going to infinity in
the limit. The answer to this part is
then,
[Return to Problems]
(b)
First, note that the limit going to negative infinity here
isn’t a violation (necessarily) of the fact that we can’t plug negative
numbers into the logarithm. The real
issue is whether or not the argument of the log will be negative or not.
Using the techniques from earlier in this section we can
see that,
and let’s also not that for negative numbers
(which we can assume we’ve got since we’re going to minus infinity in the
limit) the denominator will always be positive and so the quotient will also
always be positive. Therefore, not
only does the argument go to zero, it goes to zero from the right. This is exactly what we need to do this
limit.
So, the answer
here is,
[Return to Problems]
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As a final set of examples let’s take a look at some limits
involving inverse tangents.
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Example 7 Evaluate
each of the following limits.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
Solution
The first two parts here are really just the basic limits
involving inverse tangents and can easily be found by examining the following
sketch of inverse tangents. The
remaining two parts are more involved but as with the exponential and
logarithm limits really just refer back to the first two parts as we’ll see.
(a)
As noted above all we really need to do here is look at
the graph of the inverse tangent.
Doing this shows us that we have the following value of the limit.
[Return to
Problems]
(b)
Again, not much
to do here other than examine the graph of the inverse tangent.
[Return to Problems]
(c)
Okay, in part (a) above we saw that if the argument of the
inverse tangent function (the stuff inside the parenthesis) goes to plus
infinity then we know the value of the limit.
In this case (using the techniques from the previous section) we have,
So, this limit
is,
[Return to Problems]
(d)
Even though this limit is not a limit at infinity we’re
still looking at the same basic idea here.
We’ll use part (b) from above as a guide for this limit. We know from the Infinite Limits section that
we have the following limit for the argument of this inverse tangent,
So, since the
argument goes to minus infinity in the limit we know that this limit must be,
[Return to Problems]
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To see a precise and mathematical definition of this kind of
limit see the The Definition of the Limit section
at the end of this chapter.