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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.

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

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

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

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Example:

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provided that the second limit exists Example: Easier:

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Example: limit does not exist. Example:

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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 Example: convert the expression into a fraction by rationalizing

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Example:

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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 = L 5.

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Example: l n y = l n[(1 + sin 4x) cot x ] = cot x l n(1 + sin 4x) Find Example: x x = (e l n x ) x = e x l n x

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Example: Find

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Example:

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Limits and Their Properties 1 Copyright © Cengage Learning. All rights reserved.

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