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Guillaume De l'Hôpital 1661 - 1704 8.7 day 1 LHôpitals Rule Actually, LHôpitals Rule was developed by his teacher Johann Bernoulli. De lHôpital paid Bernoulli for private lessons, and then published the first Calculus book based on those lessons. Greg Kelly, Hanford High School, Richland, Washington

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Johann Bernoulli 1667 - 1748 8.7 day 1 LHôpitals Rule

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

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

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As becomes:

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As becomes:

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LHôpitals Rule: If is indeterminate, then:

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We can confirm LHôpitals rule by working backwards, and using the definition of derivative:

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Example: If its no longer indeterminate, then STOP! If we try to continue with LHôpitals rule: which is wrong, wrong, wrong!

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

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LHôpitals 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.

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This approaches We already know that but if we want to use LHôpitals rule: = =

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

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LHôpital again. = = =

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

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Indeterminate Forms: LHôpital applied Example: = = = = = =

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Rules for Differentiation Colorado National Monument Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

Rules for Differentiation Colorado National Monument Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

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