Proof for for convergence of $\int_{-∞}^0\frac{\sqrt{\left|x\right|}}{x^2+x+1}dx$ I want to show that the following integral converges. However I am troubled finding an estimate for this term which is sufficient for the proof. Could I use L'Hospital on this one?
$$\int_{-∞}^0\frac{\sqrt{\left|x\right|}}{x^2+x+1}dx$$

Note that $x^2+x+1=\frac{1}{2}(x^2+1)+\frac{1}{2}(x+1)^2\geq\frac{1}{2}(x^2+1)$, and thus $$\int_{-\infty}^0\frac{\sqrt{|x|}}{x^2+x+1}\leq 2\int_{-\infty}^0\frac{\sqrt{|x|}}{x^2+1}.$$
Now, the integral from $-1$ to $0$ converges, and for $|x|>1$ we have $\frac{\sqrt{|x|}}{x^2+1}<\frac{1}{|x|^\frac{3}{2}}$. And we know that the integral
$$\int_{-\infty}^{-1}\frac{1}{|x|^\frac{3}{2}}=\Big[\frac{1}{2}\frac{1}{\sqrt{|x|}}\Big]_{-\infty}^{-1}=\frac{1}{2}<\infty$$ converges.
However, I don't understand what you mean by L'Hospital in this connection.
