We provide a proof of this theorem in the special case when
and
are all continuous over an open interval containing
In that case, since
and
and
are continuous at
it follows that
Therefore,
Note that L’Hôpital’s rule states we can calculate the limit of a quotient
by considering the limit of the quotient of the derivatives
It is important to realize that we are not calculating the derivative of the quotient
□
Applying l’hôpital’s rule (0/0 case)
Evaluate each of the following limits by applying L’Hôpital’s rule.
Since the numerator
and the denominator
we can apply L’Hôpital’s rule to evaluate this limit. We have
As
the numerator
and the denominator
Therefore, we can apply L’Hôpital’s rule. We obtain
As
the numerator
and the denominator
Therefore, we can apply L’Hôpital’s rule. We obtain
As
both the numerator and denominator approach zero. Therefore, we can apply L’Hôpital’s rule. We obtain
Since the numerator and denominator of this new quotient both approach zero as
we apply L’Hôpital’s rule again. In doing so, we see that
We can also use L’Hôpital’s rule to evaluate limits of quotients
in which
and
Limits of this form are classified as
indeterminate forms of type
Again, note that we are not actually dividing
by
Since
is not a real number, that is impossible; rather,
is used to represent a quotient of limits, each of which is
or
L’hôpital’s rule
Case)
Suppose
and
are differentiable functions over an open interval containing
except possibly at
Suppose
(or
and
(or
Then,
assuming the limit on the right exists or is
or
This result also holds if the limit is infinite, if
or
or the limit is one-sided.
Applying l’hôpital’s rule
Case)
Evaluate each of the following limits by applying L’Hôpital’s rule.
Since
and
are first-degree polynomials with positive leading coefficients,
and
Therefore, we apply L’Hôpital’s rule and obtain
Note that this limit can also be calculated without invoking L’Hôpital’s rule. Earlier in the chapter we showed how to evaluate such a limit by dividing the numerator and denominator by the highest power of
in the denominator. In doing so, we saw that
L’Hôpital’s rule provides us with an alternative means of evaluating this type of limit.
Here,
and
Therefore, we can apply L’Hôpital’s rule and obtain
Now as
Therefore, the first term in the denominator is approaching zero and the second term is getting really large. In such a case, anything can happen with the product. Therefore, we cannot make any conclusion yet. To evaluate the limit, we use the definition of
to write
Now
and
so we apply L’Hôpital’s rule again. We find