Evaluate the limit: $$ \lim_{x\to0}\frac{\pi - 4\arctan{1\over 1+x}}{x} $$
I've been able to show the limit is equal to $2$ using L'Hopital's rule. After finding the derivative of the nominator the limits simply becomes: $$ \lim_{x\to0}\frac{4}{x^2 + 2x+2} = 2 $$
I'm looking for a way to find the limit without involving derivatives, but rather using some elementary methods. I've also played around with the identities involving $\arctan x$ but didn't find anything suitable.
Could someone please suggest a method to solve that problem?