1. **State the problem:** Find the limit as $x$ approaches $-5$ of the expression $$\frac{\frac{1}{5} + \frac{1}{x}}{10} + 2x.$$\n\n2. **Rewrite the expression:** The expression can be written as $$\frac{\frac{1}{5} + \frac{1}{x}}{10} + 2x = \frac{1}{10} \left( \frac{1}{5} + \frac{1}{x} \right) + 2x.$$\n\n3. **Substitute $x = -5$ directly:** Since the function is defined at $x = -5$ (no division by zero), we can substitute directly to find the limit.\n\nCalculate each part:\n$$\frac{1}{5} = 0.2,$$\n$$\frac{1}{-5} = -0.2,$$\nso $$\frac{1}{5} + \frac{1}{-5} = 0.2 - 0.2 = 0.$$\n\n4. **Evaluate the whole expression:**\n$$\frac{1}{10} \times 0 + 2 \times (-5) = 0 - 10 = -10.$$\n\n5. **Conclusion:** The limit is $$\boxed{-10}.$$