This preview shows pages 1–3. Sign up to view the full content.
Indeterminate Forms and L’Hospital’s Rule
Back in the chapter on Limits we saw methods for dealing with the following limits.
In the first limit if we plugged in
we would get 0/0 and in the second
limit if we “plugged” in infinity we would get
(
recall
that
as
x
goes to infinity a polynomial will behave in the same fashion that it’s largest
power behaves). Both of these are called
indeterminate forms
. In both of these
cases there are competing interests or rules and it’s not clear which will win out.
In the case of 0/0 we typically think of a fraction that has a numerator of zero as being
zero. However, we also tend to think of fractions in which the denominator is going
to zero as infinity or might not exist at all. Likewise, we tend to think of a fraction in
which the numerator and denominator are the same as one. So, which will win out?
Or will neither win out and they all “cancel out” and the limit will reach some other
value?
In the case of
we have a similar set of problems. If the numerator
of a fraction is going to infinity we tend to think of the whole fraction going to
infinity. Also if the denominator is going to infinity we tend to think of the fraction as
going to zero. We also have the case of a fraction in which the numerator and
denominator are the same (ignoring the minus sign) and so we might get 1. Again,
it’s not clear which of these will win out, if any of them will win out.
With the second limit there is the further problem that infinity isn’t really a number
and so we really shouldn’t even treat it like a number. Much of the time it simply
won’t behave as we would expect it to if it was a number. To look a little more into
this check out the
Types of Infinity
section in the Extras chapter at the end of this
document.
This is the problem with indeterminate forms. It’s just not clear what is happening in
the limit. There are other types of indeterminate forms as well. Some other types are,
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThese all have competing interests or rules that tell us what should happen and it’s just
not clear which, if any, of the interests or rules will win out. The topic of this section
is how to deal with these kinds of limits.
As already pointed out we do know how to deal with some kinds of indeterminate
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '08
 sc
 Limits

Click to edit the document details