Limits practice question 26

Each root grows without bound, so the difference is an indeterminate form. What turns a difference of roots into a quotient, and what changes because $x$ is negative?

Multiply and divide by the conjugate $\sqrt{x^{2} + 5x} + \sqrt{x^{2} - 3x}$. The numerator becomes $8x$.

To divide the denominator by $x$, use $\sqrt{x^{2}} = |x| = -x$ for $x < 0$: $\sqrt{x^{2} + 5x} = -x\sqrt{1 + \tfrac{5}{x}}$.