Finding $a, b\in\mathbb{R}$ so that $\lim_{x\to \infty} \sqrt{ax^2 + bx} - \sqrt{x^2 + x + 1} = 1$ Find the values of real constants $a$ and $b$ such that 
$$\lim_{x\to \infty} \sqrt{ax^2 + bx} - \sqrt{x^2 + x + 1} = 1$$
If I obtained $a = 1$  and $b=0$, is that the correct answer? 
Or is $b$ actually some other value? If so, then why? 

Hint: $$\dfrac{ax^2 + bx - x^2 - x - 1}{\sqrt{ax^2 + bx} + \sqrt{x^2 + x + 1}} = \dfrac{(a-1)x + (b-1) - \frac{1}{x}}{\sqrt{a + \frac{b}{x}} + \sqrt{1 + \frac{1}{x} + \frac{1}{x^2}}}.$$
