Intermediate Forms and L’Hospital Rule (also L’Hopital and Bernoulli Rule)

1. 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'Hôpitalthanks 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 If exists, where ,the limit is said to be indeterminate. 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.

2. I Indeterminate Form • You have previously studied limits with the indeterminate form Example: Example: We used a geometric argument to show that: Example: Example: Some limits can be recognized as a derivative

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

4. Not all forms are like those. with the knowledge given to you • If you can find by now you get a doctorate at 17 and get to quit the school right now. L’Hospital rule for indeterminate form Let f  and g  be real functions which are continuous on the closed interval [a, b]  and differentiable on the open interval (a, b)  . Suppose that . Then: provided that the second limit exists.

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6. Example: Example: Example: Example: Example:

7. II Indeterminate Form Suppose that instead of , we have that   and   as . Then: Corollary for indeterminate form provided that the second limit exists = Example: Easier:

8. Example: = limit does not exist. = Example: Example: = Example:

9. Example: Rule does not help in this situation. It is a pure pain. In the situations like this one divide in you mind denominator and numerator with highest exponent of III Indeterminate Form 0 ∙ ∞ use algebra to convert the expression to a fraction (0 ), and then apply L'Hopital's Rule Example:

10. Example: Example: Example: Example: Example:

11. IV Indeterminate Form A limit problem that leads to one of the expressions is called an indeterminate form of type • 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 Indeterminate forms of the type can sometimes be evaluated by combining the terms and manipulating the result to produce an indeterminate form of type Example: convert the expression into a fraction by rationalizing

12. Example: Example: Example: Example:

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

14. Example: lny = ln[(1 + sin 4x)cot x] = cot xln(1 + sin 4x) Example: Find xx = (elnx)x = exlnx

15. Example: Find

16. Example: Find