1. **State the problem:** We need to find the limit as $x$ approaches 5 of the expression $$\frac{\frac{1}{5+x}}{10+2x}.$$\n\n2. **Rewrite the expression:** The expression can be simplified by dividing the numerator by the denominator:\n$$\frac{\frac{1}{5+x}}{10+2x} = \frac{1}{5+x} \times \frac{1}{10+2x} = \frac{1}{(5+x)(10+2x)}.$$\n\n3. **Evaluate the limit:** Since the function is continuous at $x=5$ (no division by zero or undefined terms), we can substitute $x=5$ directly:\n$$\lim_{x \to 5} \frac{1}{(5+x)(10+2x)} = \frac{1}{(5+5)(10+2\times5)} = \frac{1}{10 \times 20} = \frac{1}{200}.$$\n\n4. **Conclusion:** The limit is $$\boxed{\frac{1}{200}}.$$