1. **State the problem:** We want to evaluate the limit $$\lim_{x \to 25} \frac{\sqrt{x} - 5}{x - 25}$$. 2. **Recognize the indeterminate form:** Direct substitution gives $$\frac{\sqrt{25} - 5}{25 - 25} = \frac{5 - 5}{0} = \frac{0}{0}$$ which is indeterminate, so we must simplify. 3. **Rewrite the expression:** Multiply numerator and denominator by the conjugate of the numerator to simplify. $$\frac{\sqrt{x} - 5}{x - 25} \times \frac{\sqrt{x} + 5}{\sqrt{x} + 5} = \frac{(\sqrt{x} - 5)(\sqrt{x} + 5)}{(x - 25)(\sqrt{x} + 5)}$$ 4. **Simplify numerator:** Use difference of squares: $$ (\sqrt{x})^2 - 5^2 = x - 25 $$ So the expression becomes: $$ \frac{x - 25}{(x - 25)(\sqrt{x} + 5)} $$ 5. **Cancel common factors:** Since $$x \neq 25$$ near the limit, cancel $$x - 25$$: $$ \frac{1}{\sqrt{x} + 5} $$ 6. **Evaluate the limit:** Substitute $$x = 25$$: $$ \frac{1}{\sqrt{25} + 5} = \frac{1}{5 + 5} = \frac{1}{10} $$ **Final answer:** $$ \boxed{\frac{1}{10}} $$