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June 22nd, 2012, 01:18 PM   #1
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Applying L'Hopital's Rule

It starts in the form 0/0 so it is of the right form to apply L'Hopital's Rule but the trig numerator I think needs a manipulation and so does the denominator where the ln is ln(x+1)^2 which goes to 2ln(x+1) but I am a bit stuck any ideas?
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June 22nd, 2012, 04:48 PM   #2
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Re: Applying L'Hopital's Rule

You can use L'Hopital. But, it takes about 3 applications.

You can write it as

For the left side, applying L'H 3 times brings us to:

Now, letting x=0, gives

For the right side, after applying L'H 3 times, we get:

Letting x=0, gives

Putting them together, gives

This rather daunting to do all of that differentiation. There is more than likely a better and more clever way.

Someone will come along with a sharp method without L'H.
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June 22nd, 2012, 04:53 PM   #3
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Re: Applying L'Hopital's Rule

thanks man going through your steps to see where I messed up!
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June 22nd, 2012, 09:01 PM   #4
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Re: Applying L'Hopital's Rule

Because x is approaching 0, you can immediately use the well-known series:

For very small x, you can discard higher powers, so the numerator is
so the denominator is
and then
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