1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{\sqrt{x^2 + 100} - 10}{x^2}.$$\n\n2. **Recall the formula and approach:** When direct substitution leads to an indeterminate form like $\frac{0}{0}$, we can use algebraic manipulation such as multiplying by the conjugate to simplify the expression.\n\n3. **Check direct substitution:** Substitute $x=0$:\n$$\frac{\sqrt{0^2 + 100} - 10}{0^2} = \frac{10 - 10}{0} = \frac{0}{0},$$ which is indeterminate.\n\n4. **Multiply numerator and denominator by the conjugate:**\n$$\frac{\sqrt{x^2 + 100} - 10}{x^2} \times \frac{\sqrt{x^2 + 100} + 10}{\sqrt{x^2 + 100} + 10} = \frac{(x^2 + 100) - 100}{x^2 (\sqrt{x^2 + 100} + 10)} = \frac{x^2}{x^2 (\sqrt{x^2 + 100} + 10)}.$$\n\n5. **Simplify the expression:**\n$$\frac{x^2}{x^2 (\sqrt{x^2 + 100} + 10)} = \frac{1}{\sqrt{x^2 + 100} + 10}.$$\n\n6. **Evaluate the limit as $x \to 0$:**\n$$\lim_{x \to 0} \frac{1}{\sqrt{x^2 + 100} + 10} = \frac{1}{\sqrt{0 + 100} + 10} = \frac{1}{10 + 10} = \frac{1}{20}.$$\n\n**Final answer:** $$\boxed{\frac{1}{20}}.$$