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Learn the concept of L’Hôpital's Rule, its application in evaluating indeterminate forms, and practical examples. Understand how to utilize this rule in calculus to solve complex mathematical problems effectively.
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8.2 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. Guillaume De l'Hôpital 1661 - 1704
If we try to evaluate this by direct substitution, we get: Consider: 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:
L’Hôpital’s Rule: If is indeterminate, then:
We can confirm L’Hôpital’s rule by working backwards, and using the definition of derivative:
Example: If it’s no longer indeterminate, then STOP! If we try to continue with L’Hôpital’s rule: which is wrong, wrong, wrong!
not 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.)
L’Hôpital’s rule can be used to evaluate other indeterminate forms besides .
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.
Here is the standard list of the seven indeterminate forms: These expressions can be evaluated “as is” using L’Hopital’s Rule. These expressions have to be changed to a fraction before using L’Hopital’s Rule!!! There are other indeterminate forms using complex numbers, but those are beyond the scope of this class.
This approaches This approaches We already know that but if we want to use L’Hôpital’s rule:
This is indeterminate form Now it is in the form L’Hôpital’s rule applied once. Fractions cleared. Still If we find a common denominator and subtract, we get:
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: Evaluating these forms requires a mathematical trick to change the expression into a fraction.
L’Hôpital applied Indeterminate Forms: Example:
