



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 
