# Calc08_1 - 8.1 L'Hpital's Rule Actually L'Hpital's Rule was...

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Guillaume De l'Hôpital 1661 - 1704 8.1: 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. Greg Kelly, Hanford High School, Richland, Washington

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Johann Bernoulli 1667 - 1748 8.1: L’Hôpital’s Rule
Zero divided by zero can not be evaluated, and is an example of indeterminate form . 2 2 4 lim 2 x x x - - Consider: If we try to evaluate this by direct substitution, we get: 0 0 In this case, we can evaluate this limit by factoring and canceling: 2 2 4 lim 2 x x x - - ( 29 ( 29 2 2 2 lim 2 x x x x + - = - ( 29 2 lim 2 x x = + 4 =

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2 2 4 lim 2 x x x - = - The limit is the ratio of the numerator over the denominator as x approaches 2. 2 4 x - 2 x - ( 29 ( 29 lim x a f x g x
2 2 4 lim 2 x x x - = - ( 29 ( 29 lim x a f x g x ( 29 f x ( 29 g x ( 29 ( 29 f x g x As 2 x becomes:

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2 2 4 lim 2 x x x - = - ( 29 ( 29 lim x a f x g x As 2 x ( 29 ( 29 f x g x becomes: df dg df dg dx dx df x dg d =
2 2 4 lim 2 x x x - = - ( 29 ( 29 lim x a f x g x ( 29 ( 29 2 2 4

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