How can one find $\lim_{x \to \pm\infty}\frac{2x+1}{\sqrt{x^2+x+1}}$ without graphing? I'm trying to find both of these:
$$\lim_{x \to +\infty}\frac{2x+1}{\sqrt{x^2+x+1}}$$
$$\lim_{x \to -\infty}\frac{2x+1}{\sqrt{x^2+x+1}}$$

I know that they end up being $2$ and $-2$ by graphing, but what process can I use to find them without graphing (in the simplest way possible)?

$$\frac{2x+1}{\sqrt{x^2+x+1}}=\frac{x\left(2+\dfrac{1}{x}\right)}{\sqrt{x^2\left(1+\dfrac{1}{x}+\dfrac{1}{x^2}\right)}}=\frac{x\left(2+\dfrac{1}{x}\right)}{|x|\sqrt{1+\dfrac{1}{x}+\dfrac{1}{x^2}}}=\cdots$$
