Some $\frac{0}{0}$ limits involve square roots, and plain factoring does not reveal the common factor. The trick from Simplifying and Rationalizing Radicals helps: multiply by a conjugate. It works for the same reason as Limits by Factoring and Canceling: it exposes a factor we can cancel.
Key idea
For $\sqrt{A} - B$, multiply top and bottom by the conjugate $\sqrt{A} + B$. The product $(\sqrt{A} - B)(\sqrt{A} + B) = A - B^2$ has no root, so the hidden factor appears and can be canceled.
Why it works
Multiplying by $\frac{\sqrt{A} + B}{\sqrt{A} + B}$ is multiplying by 1 (wherever the conjugate is not 0), so the expression keeps its value. The difference of squares $(u - v)(u + v) = u^2 - v^2$ turns $\sqrt{x + 9} - 3$ into $(x + 9) - 9 = x$. Now the $x$ on top can cancel the $x$ on the bottom, which was the source of the $\frac{0}{0}$.
Worked example: Root in the numerator
Find $\displaystyle\lim_{x \to 0} \frac{\sqrt{x + 9} - 3}{x}$.
- Substituting gives $\frac{3 - 3}{0} = \frac{0}{0}$.
- Multiply top and bottom by $\sqrt{x + 9} + 3$.
- Top: $(\sqrt{x + 9})^2 - 3^2 = x + 9 - 9 = x$. Bottom: $x(\sqrt{x + 9} + 3)$.
- Cancel $x$: $\frac{1}{\sqrt{x + 9} + 3}$.
- Substitute: $\frac{1}{3 + 3} = \frac{1}{6}$.
Answer: $\frac{1}{6}$
Worked example: Root in the denominator
Find $\displaystyle\lim_{x \to 4} \frac{x - 4}{\sqrt{x} - 2}$.
- $\frac{0}{0}$. Multiply by $\frac{\sqrt{x} + 2}{\sqrt{x} + 2}$.
- Bottom: $x - 4$, which cancels with the top.
- What remains is $\sqrt{x} + 2$, which tends to $4$.
Answer: $4$ (another way: $x - 4 = (\sqrt{x} - 2)(\sqrt{x} + 2)$ factors directly)
Worked example: A shifted root
Find $\displaystyle\lim_{x \to 1} \frac{\sqrt{x + 3} - 2}{x - 1}$.
- $\frac{0}{0}$. Multiply by $\sqrt{x + 3} + 2$.
- Top: $(x + 3) - 4 = x - 1$, which cancels with the bottom.
- Left: $\frac{1}{\sqrt{x + 3} + 2} \to \frac{1}{4}$.
Answer: $\frac{1}{4}$
- Reach for a conjugate when substitution gives $\frac{0}{0}$ and a square root sits next to a plus or minus sign.
- Multiply by the same root expression with the opposite sign in the middle.
- Multiply out only the part with the root; keep the other part factored, because that is where the cancellation happens.
- After canceling, substitute. If you still get $\frac{0}{0}$, look for a second common factor.
Common mistake
Multiplying only the numerator by the conjugate changes the value of the expression. Multiply top and bottom, and do not expand the bottom: leave it factored so the cancellation is easy to see. Check: your simplified expression should agree with the original at a test point like $x = 0.01$.
The first example is secretly a slope: it measures how fast $\sqrt{x}$ grows at $x = 9$. You will meet it again when we differentiate $\sqrt{x}$.