1. **State the problem:** Find the limit $$\lim_{x \to 0} \left( \frac{x}{x+1} - 1 \right).\n\n2. **Rewrite the expression:** \n$$\frac{x}{x+1} - 1 = \frac{x}{x+1} - \frac{x+1}{x+1} = \frac{x - (x+1)}{x+1}.$$\n\n3. **Simplify the numerator:** \n$$x - (x+1) = x - x - 1 = -1.$$\n\n4. **Substitute back:** \n$$\frac{-1}{x+1}.$$\n\n5. **Evaluate the limit:** \nAs $x \to 0$, $x+1 \to 1$, so \n$$\lim_{x \to 0} \frac{-1}{x+1} = \frac{-1}{1} = -1.$$\n\n**Final answer:** $$-1.$$