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?

Hint:
$$ \arctan(x)\pm\arctan(y) = \arctan(z) $$
where $z$ is:
$$ z = \frac{x\pm y}{1\mp xy} $$
And knowing the fact that:
$$ \lim_{x\to0} \frac{\arctan(f(x))}{f(x)} = 1 $$
If $ f(x) = 0 $ as $x\to0$.
This can be proven by the L'Hospital's rule:
$$ \lim_{x\to0} \frac{\arctan(f(x))}{f(x)} = \lim_{x\to0} \frac{\frac{1}{1+f^2(x)}f'(x)}{f'(x)} = \lim_{x\to0} \frac{1}{1+f^2(x)} = 1. $$
