1. **State the problem:** Find the limit $$\lim_{x \to 5} \frac{\frac{1}{5+x}}{10+2x}$$. 2. **Rewrite the expression:** The expression is a complex fraction. We can rewrite it as $$\frac{\frac{1}{5+x}}{10+2x} = \frac{1}{5+x} \times \frac{1}{10+2x} = \frac{1}{(5+x)(10+2x)}.$$ 3. **Evaluate the limit by direct substitution:** Since the function is continuous at $x=5$, substitute $x=5$: $$ (5+5)(10 + 2 \times 5) = 10 \times (10 + 10) = 10 \times 20 = 200.$$ 4. **Calculate the limit:** $$\lim_{x \to 5} \frac{1}{(5+x)(10+2x)} = \frac{1}{200}.$$ 5. **Conclusion:** The limit is $$\boxed{\frac{1}{200}}.$$