Determinate set $A\subseteq\mathbb{R}$ so that for every $a\in A$ and every $x\in\mathbb{R}$ the condition $ax^2+x+3\ge0$ is valid Quadratic function is always greater than $0$ if
$$a>0$$ and $$D=0$$
Solving for $D$ we have
$$1-12a=0\Rightarrow a=\frac{1}{12}$$
So, $$a\in[\frac{1}{12},+\infty)$$
Is this the only condition to check?
For found $a\in \mathbb{A}$ how to evaluate 
$$\lim\limits_{x\to\infty}(x+1-\sqrt{ax^2+x+3})$$

$ax^2+x+3\ge 0$ for every $x\in\mathbb R$ if and only if $a\gt 0$ and $D=1-12a\color{red}{\le} 0$, i.e. $a\color{red}{\ge} \frac{1}{12}$. 
For the limit, you can have
$$\begin{align}\lim_{x\to\infty}\left(x+1-\sqrt{ax^2+x+3}\right)&=\lim_{x\to\infty}\frac{(x+1-\sqrt{ax^2+x+3})(x+1+\sqrt{ax^2+x+3})}{x+1+\sqrt{ax^2+x+3}}\\&=\lim_{x\to\infty}\frac{(x+1)^2-(ax^2+x+3)}{x+1+\sqrt{ax^2+x+3}}\\&=\lim_{x\to\infty}\frac{(1-a)x^2+x-2}{x+1+\sqrt{ax^2+x+3}}\\&=\lim_{x\to\infty}\frac{(1-a)x+1-\frac 2x}{1+\frac 1x+\sqrt{a+\frac 1x+\frac{3}{x^2}}}\end{align}$$


*

*If $1-a\gt 0$, i.e. $a\lt 1$, then the limit is $+\infty$.

*If $1-a=0$, i.e. $a=1$, then the limit is $\frac 12$.

*If $1-a\lt 0$, i.e. $a\gt 1$, then the limit is $-\infty$.
