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Infinite Zeros
L'Hopital's Rule

 

 Sometimes when we do an integral, we wind up with something like ...

 

 

 
 These turn out to be infinity.
 Sometimes, however, e wind up with something like this ...
 

 
 Up to now, when we got something like this we were stuck,
 but thanks to old L'Hopital, now we can deal with them.
 L'Hopital's rule says that if a problem gives us ...
 

 
 We can take the derivatives of F(X) and G(X)
 and solve for the nasty X value.
 Whatever we get as an answer is also the answer
 for the original "pre derivatives" function.
 
 If we have ...
 

 
 We can take the derivatives of 3X and 2X and get ...
 

 
 So that means ...
 

 
 OK, the deal to remember with these
 is that you take the derivative of the numerator
 and the derivative of the denominator
 separately ...
 
 DO NOT USE THE QUOTIENT RULE!!!!
 
 Sometimes to get an answer, 
 we need to take derivatives more than once.
 
 Say we have ...
 

 
 Step 1 - Find the derivative of the numerator
 and the derivative of the denominator ...
 

 
 Step 2
 
 

 
 Step 3 - Find the derivative of the numerator 
 and the derivative of the denominator again ...
 

 
 Did you notice that when the greatest exponent in the numerator
 is the same as the greatest exponent in the denominator,
 the limit is the coefficients!
 (We actually did this back in graphing with derivatives.)
 
 Another Example:
 

 
 Hey wait a minute!
 That one's not 0/0 or / it's 0 - !      
 What's up?
 
 Well 0 can be arranged into either 0/0 or /.
 
 Oh yeah? Let's see ...
 

 
 The sign on an infinity really doesn't matter for this,
 but let's do a bit of rewriting ...
 

 
 Now take the derivatives separately ...
 

 
 Now we simplify ...
 

 
 To Recap ...
 
 L'Hopital's rule may be used when we have ...
 

 
 But it is of no use for ...
 

 
 What about 0 ?
 
 OK, that was a trick ...
 You had that one in algebra!
 
 Anything to the zero power ... (you know the words)
 

   copyright 2005 Bruce Kirkpatrick

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