1. **State the problem:** Evaluate the limit $$\lim_{x \to 5} \frac{\sqrt{5} - \sqrt{x}}{x - 5}$$. 2. **Recall the formula and rules:** Direct substitution gives $$\frac{\sqrt{5} - \sqrt{5}}{5 - 5} = \frac{0}{0}$$ which is indeterminate. We use algebraic manipulation to simplify. 3. **Multiply numerator and denominator by the conjugate:** $$\frac{\sqrt{5} - \sqrt{x}}{x - 5} \times \frac{\sqrt{5} + \sqrt{x}}{\sqrt{5} + \sqrt{x}} = \frac{(\sqrt{5})^2 - (\sqrt{x})^2}{(x - 5)(\sqrt{5} + \sqrt{x})} = \frac{5 - x}{(x - 5)(\sqrt{5} + \sqrt{x})}$$ 4. **Simplify the numerator and denominator:** Note that $$5 - x = -(x - 5)$$, so $$\frac{5 - x}{(x - 5)(\sqrt{5} + \sqrt{x})} = \frac{-(x - 5)}{(x - 5)(\sqrt{5} + \sqrt{x})} = \frac{-1}{\sqrt{5} + \sqrt{x}}$$ 5. **Evaluate the limit by direct substitution:** $$\lim_{x \to 5} \frac{-1}{\sqrt{5} + \sqrt{x}} = \frac{-1}{\sqrt{5} + \sqrt{5}} = \frac{-1}{2\sqrt{5}}$$ 6. **Final answer:** $$\boxed{\frac{-1}{2\sqrt{5}}}$$