Key property: $(a+b)(a-b) = a^2 - b^2$

This eliminates the square root!

Standard Approach

1. Identify the expression with the square root
2. Multiply numerator and denominator by the conjugate
3. Simplify using $(a+b)(a-b) = a^2 - b^2$
4. Cancel common factors
5. Evaluate the limit

Example 1: Square Root in Numerator

$\lim_{x \to 0} \frac{\sqrt{x+4} - 2}{x}$

Direct substitution: $\frac{0}{0}$ ✗

Multiply by conjugate $\sqrt{x+4} + 2$:

$\lim_{x \to 0} \frac{\sqrt{x+4} - 2}{x} \cdot \frac{\sqrt{x+4} + 2}{\sqrt{x+4} + 2}$

Simplify numerator: $(\sqrt{x+4} - 2)(\sqrt{x+4} + 2) = (x+4) - 4 = x$

$\lim_{x \to 0} \frac{x}{x(\sqrt{x+4} + 2)}$

Cancel $x$: $\lim_{x \to 0} \frac{1}{\sqrt{x+4} + 2}$

Direct substitution: $= \frac{1}{\sqrt{4} + 2} = \frac{1}{4}$

Example 2: Square Root in Denominator

$\lim_{h \to 0} \frac{h}{\sqrt{h+1} - 1}$

Multiply by conjugate of denominator:

$\lim_{h \to 0} \frac{h}{\sqrt{h+1} - 1} \cdot \frac{\sqrt{h+1} + 1}{\sqrt{h+1} + 1}$

$= \lim_{h \to 0} \frac{h(\sqrt{h+1} + 1)}{h} = \lim_{h \to 0} (\sqrt{h+1} + 1) = 2$

Remember

Conjugates turn $a^2 - b^2$ differences into factored forms, eliminating radicals!