<s>[INST] What are the values of the parameters $a,b \in \mathbb{R}$ such that $\lim\limits_{x \to \infty}(\sqrt{x^2+x+1}+\sqrt{x^2+2x+2}-ax-b)=0.$ I have to find the values of $a$ and $b$ (with $a,b \in \mathbb{R}$) such that the following is true:
$$\lim\limits_{x \to \infty} (\sqrt{x^2+x+1} + \sqrt{x^2+2x+2}-ax-b)=0$$
This is what I did:
$$\lim\limits_{x \to \infty} (\sqrt{x^2+x+1} + \sqrt{x^2+2x+2}-ax-b)=0$$
$$\lim\limits_{x \to \infty} \bigg (x\sqrt{1+\dfrac{1}{x}+\dfrac{1}{x^2}} + x\sqrt{1+\dfrac{2}{x}+\dfrac{2}{x^2}}-ax-b \bigg )=0$$
$$\lim\limits_{x \to \infty} x\bigg (\sqrt{1+\dfrac{1}{x}+\dfrac{1}{x^2}} + \sqrt{1+\dfrac{2}{x}+\dfrac{2}{x^2}}-a-\dfrac{b}{x} \bigg )=0$$
So, we'd have something like:
$$\infty \cdot(2-a) = 0$$
$(*)$If $a \in (-\infty, 2)$ $\Rightarrow$ $(2-a) > 0$, which means:
$$\infty \cdot(2-a) = \infty$$
$(*)$If $a\in (2, +\infty)$ $\Rightarrow$ $(2-a) < 0$, which means:
$$\infty \cdot(2-a) = -\infty$$
So the only option left for the limit to have any chance of being true is:
$$a=2$$
which would result in the indeterminate form:
$$\infty \cdot (2-a) = \infty \cdot 0$$
And now that we have $a=2$, we need to find the value of $b$ for the limit to be true. Substituting $a$ with $2$ in the initial limit we get:
$$\lim\limits_{x \to \infty} (\sqrt{x^2+x+1} + \sqrt{x^2+2x+2}-2x-b)=0$$
Since $b$ is a constant we can pull it out of the limit and get:
$$b=\lim\limits_{x \to \infty} (\sqrt{x^2+x+1} + \sqrt{x^2+2x+2}-2x)$$
And this is where I got stuck. I tried a bunch of methods and tricks for finding this limit and I got nowhere. That leads me to think that either I made some mistake/mistakes along the way, or I simply don't know how to solve this limit. So, how can I find $b$?
 [/INST] Hint: Intuitively, for large $x$, we have $\sqrt{x^2+x+1} = \sqrt{(x+\tfrac{1}{2})^2+\tfrac{3}{4}} \approx x+\tfrac{1}{2}$ and $\sqrt{x^2+2x+2} = \sqrt{(x+1)^2+1} \approx x+1$.  But how can we make these observations rigorous? 
Using the identity $\sqrt{a}-\sqrt{b} = \dfrac{a-b}{\sqrt{a}+\sqrt{b}}$, we get that$$\sqrt{x^2+x+1} - (x+\tfrac{1}{2}) = \dfrac{(x^2+x+1)-(x+\tfrac{1}{2})^2}{\sqrt{x^2+x+1} + (x+\tfrac{1}{2})} = \dfrac{\tfrac{3}{4}}{\sqrt{x^2+x+1} + (x+\tfrac{1}{2})}$$  and $$\sqrt{x^2+2x+2} - (x+1) = \dfrac{(x^2+2x+2)-(x+1)^2}{\sqrt{x^2+2x+2} + (x+1)} = \dfrac{1}{\sqrt{x^2+2x+2} + (x+1)}.$$ Both of these expressions tend to $0$ as $x \to \infty$. Do you see how to use these results to solve the problem?
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