Intermediate Forms and LHospital Rule also LHopital and

Intermediate Forms and L’Hospital Rule (also L’Hopital and Bernoulli Rule) Historically, this result first appeared in L'Hôpital's 1696 treatise, which was the first textbook on differential calculus. Within the book, L'Hôpital thanks the Bernoulli brothers for their assistance and their discoveries. An earlier letter by John Bernoulli gives both the rule and its proof, so it seems likely that Bernoulli discovered the rule. Definition: Indeterminate Limit/Form The following expressions are indeterminate forms: These expressions are called indeterminate because you cannot determine their exact value in the indeterminate form. However, it is still possible to solve these in many cases due to L'Hôpital's rule.

Ex: We used a geometric argument to show that: Ex: Some limits can be recognized as a derivative

Recognizing a given limit as a derivative (!!!!!!) Ex: Ex: Tricky, isn’t it? A lot of grey cells needed.

If you can find by now you get a doctorate at 17 and get to quit the school right now. provided that the second limit exists.

Ex: Ex:

Ex:

Ex:

Example: Easier: provided that the second limit exists

Ex: limit does not exist. Ex:

Ex: Conclusion: L’Hospital is not always the most powerful tool

and then apply L'Hopital's Rule Ex:

Ex: Ex:

A limit problem that leads to one of the expressions Such limits are indeterminate because the two terms exert conflicting influences on the expression; one pushes it in the positive direction and the other pushes it in the negative direction Ex: convert the expression into a fraction by rationalizing

Ex: Ex:

Several indeterminate forms arise from the limit These indeterminate forms can sometimes be evaluated as follows: 1. 2. 3. Find the limit of both sides 4. If 5. =L

Ex: Find

Ex: Find

Ex: Find