Evaluating $ \lim_{x\to +\infty} x\left(\frac{\pi}{4} - \arctan\left(\frac{x}{x+1}\right)\right) $ 
$$ \lim_{x\to +\infty} x\left(\frac{\pi}{4} - \arctan\left(\frac{x}{x+1}\right)\right) $$

I tried to do this with some kind of substitution but failed miserably. Any hints or help?

Substitute $y:=\frac{1}{x+1}$ to rewrite your limit as$$\lim_{y\to0}(1-y)\frac{(\arctan 1-\arctan(1-y))}{y}=\lim_{y\to0}(1-y)\cdot\arctan^\prime1=\frac12.$$Or if we take @DinnoKoluh's approach,$$\arctan1-\arctan\frac{x}{x+1}=\arctan\frac{1}{2x+1}\approx\frac{1}{2x}.$$
