1. **State the problem:** Evaluate the limit $$\lim_{x \to 100} \frac{\sqrt{x} - 10}{x - 100}$$. 2. **Recall the formula and rules:** Direct substitution gives $$\frac{\sqrt{100} - 10}{100 - 100} = \frac{10 - 10}{0} = \frac{0}{0}$$, which is an indeterminate form. We need to simplify the expression. 3. **Simplify the expression:** Multiply numerator and denominator by the conjugate of the numerator: $$\frac{\sqrt{x} - 10}{x - 100} \times \frac{\sqrt{x} + 10}{\sqrt{x} + 10} = \frac{(\sqrt{x} - 10)(\sqrt{x} + 10)}{(x - 100)(\sqrt{x} + 10)} = \frac{x - 100}{(x - 100)(\sqrt{x} + 10)}$$. 4. **Cancel common factors:** The factor $(x - 100)$ cancels out: $$\frac{1}{\sqrt{x} + 10}$$. 5. **Evaluate the limit:** Substitute $x = 100$: $$\frac{1}{\sqrt{100} + 10} = \frac{1}{10 + 10} = \frac{1}{20}$$. **Final answer:** $$\lim_{x \to 100} \frac{\sqrt{x} - 10}{x - 100} = \frac{1}{20}$$.