7.7 L’Hôpital’s Rule Guillaume De l'Hôpital 1661 - 1704

7.7 L’Hôpital’s Rule Actually, L’Hôpital’s Rule was developed by his teacher Johann Bernoulli. De l’Hôpital paid Bernoulli for private lessons, and then published the first Calculus book based on those lessons. Johann Bernoulli 1667 - 1748

What makes an expression indeterminate? Consider: We can hold one part of the expression constant: There are conflicting trends here. The actual limit will depend on the rates at which the numerator and denominator approach infinity, so we say that an expression in this form is indeterminate.

Let’s look at another one: Consider: We can hold one part of the expression constant: Once again, we have conflicting trends, so this form is indeterminate.

Finally, here is an expression that looks like it might be indeterminate : Consider: We can hold one part of the expression constant: The limit is zero any way you look at it, so the expression is not indeterminate.

Consider: If we try to evaluate this by direct substitution, we get: Zero divided by zero can not be evaluated, and is an example of indeterminate form. In this case, we can evaluate this limit by factoring and canceling:

The limit is the ratio of the numerator over the denominator as x approaches 2. If we zoom in far enough, the curves will appear as straight lines.

As becomes:

As becomes:

L’Hôpital’s Rule: If is indeterminate, then:

Example: If it’s no longer indeterminate, then STOP! If we try to continue with L’Hôpital’s rule: which is wrong, wrong, wrong!

On the other hand, you can apply L’Hôpital’s rule as many times as necessary as long as the fraction is still indeterminate: (Rewritten in exponential form.) not

L’Hôpital’s rule can be used to evaluate other indeterminate forms besides . The following are also considered indeterminate: The first one, , can be evaluated just like . The others must be changed to fractions first.

This approaches This approaches We already know that but if we want to use L’Hôpital’s rule:

This is indeterminate form If we find a common denominator and subtract, we get: Now it is in the form L’Hôpital’s rule applied once. Fractions cleared. Still

L’Hôpital again.

Indeterminate Forms: Evaluating these forms requires a mathematical trick to change the expression into a fraction. We can then write the expression as a fraction, which allows us to use L’Hôpital’s rule. When we take the log of an exponential function, the exponent can be moved out front. Then move the limit notation outside of the log. We can take the log of the function as long as we exponentiate at the same time.

Indeterminate Forms: Example: L’Hôpital applied p

p. 537 #’s 3-51 odd

We can confirm L’Hôpital’s rule by working backwards, and using the definition of derivative: