Curve sketching practice question 7

Take the limit at each end on its own. When $x$ is large and negative, what sign does the top have, and what sign does the bottom have?

For large $|x|$, $\sqrt{4x^{2} + 1}$ behaves like $\sqrt{4x^{2}} = 2|x|$. Treat $x \to \infty$ and $x \to -\infty$ separately: $|x| = x$ in one case and $-x$ in the other.