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’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 of the two factors. The first
is clearly infinity and the second is clearly a finite number (one in this
case) and so the Facts
from the Infinite Limits section gives us the following limit,
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. Note as well that while we
wrote for the limit of the first term we are
really using the Facts
from the Infinite Limit section to do that limit.
[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, after
taking the limit of the two terms (the first is infinity and the second is a
negative, finite number) and recalling the Facts from the Infinite
Limit section we see that the limit is,
[Return to Problems]
