I have a question regarding limits

Question

Find the limit as x goes to 0:

 

(ln(1+x²)-x(arctan(x)))/(x(sin(x)-x))

 

 

At first the fraction found is 0/0, but applying L'Hopitals rule, when differentiated the answer is again 0/0, 

and when again differentiated its again 0/0... However the correct answer is 1. How?

Answer 1

You were on the right track, but gave up too soon.

 

If you apply L'Hopital's Rule FOUR (4) times, the fraction becomes:

 

   -4

---------

-4cosx + x*sinx

 

The limit as x-->0 is in fact 1