Limits practice question 13

Divide out the dominant power of $x$ from the top and the bottom. What changes when you take $x$ out of a square root and $x$ is negative?

Take $x^{2}$ out of the root: $\sqrt{9x^{2} + 4} = \sqrt{x^{2}}\,\sqrt{9 + \tfrac{4}{x^{2}}} = |x|\sqrt{9 + \tfrac{4}{x^{2}}}$. What is $|x|$ when $x$ is negative?

Check the sign before computing: for very negative $x$, the root is positive and $2x - 1$ is negative.