1. **State the problem:** Find the limit $$\lim_{x \to 0} \frac{\frac{1}{5+x} - \frac{1}{5}}{x}$$. 2. **Recall the formula:** This is a difference quotient resembling the definition of a derivative: $$\lim_{x \to 0} \frac{f(5+x) - f(5)}{x}$$ where $$f(t) = \frac{1}{t}$$. 3. **Simplify the numerator:** $$\frac{1}{5+x} - \frac{1}{5} = \frac{5 - (5+x)}{5(5+x)} = \frac{-x}{5(5+x)}$$. 4. **Substitute back into the limit:** $$\lim_{x \to 0} \frac{-x}{5(5+x) \cdot x} = \lim_{x \to 0} \frac{-x}{5(5+x)x} = \lim_{x \to 0} \frac{-1}{5(5+x)}$$. 5. **Evaluate the limit by direct substitution:** $$\frac{-1}{5(5+0)} = \frac{-1}{25}$$. **Final answer:** $$\boxed{-\frac{1}{25}}$$