Calculus 8.2 day 1

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8.2 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.
Guillaume De l'Hôpital 1661 - 1704
Greg Kelly, Hanford High School, Richland,
Washington
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8.2 day 1 LHôpitals Rule
Johann Bernoulli 1667 - 1748
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Consider
Zero divided by zero can not be evaluated, and is
an example of indeterminate form.
In this case, we can evaluate this limit by
factoring and canceling
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The limit is the ratio of the numerator over the
denominator as x approaches 2.
If we zoom in far enough, the curves will appear
as straight lines.
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As
becomes
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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
(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 others must be changed to fractions first.
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but if we want to use LHôpitals rule
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If we find a common denominator and subtract, we
get
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Indeterminate Forms
Evaluating these forms requires a mathematical
trick to change the expression into a fraction.
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Indeterminate Forms
Example
p