June 22nd, 2012, 02:18 PM  #1 
Senior Member Joined: Jan 2012 Posts: 159 Thanks: 0  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? 
June 22nd, 2012, 05:48 PM  #2 
Senior Member Joined: May 2011 Posts: 501 Thanks: 5  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. 
June 22nd, 2012, 05:53 PM  #3 
Senior Member Joined: Jan 2012 Posts: 159 Thanks: 0  Re: Applying L'Hopital's Rule
thanks man going through your steps to see where I messed up!

June 22nd, 2012, 10:01 PM  #4 
Senior Member Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3  Re: Applying L'Hopital's Rule
Because x is approaching 0, you can immediately use the wellknown series: For very small x, you can discard higher powers, so the numerator is Meanwhile so the denominator is and then 

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