1. The problem asks for the left-hand limit as $x$ approaches 0 of the expression $$\frac{x}{x - |x|}.$$\n\n2. We need to analyze the expression for values of $x$ approaching 0 from the negative side, so $x \to 0^-$.\n\n3. For $x < 0$, the absolute value function is $|x| = -x$, because the absolute value of a negative number is its positive counterpart.\n\n4. Substitute $|x| = -x$ into the expression: $$\frac{x}{x - |x|} = \frac{x}{x - (-x)} = \frac{x}{x + x} = \frac{x}{2x}.$$\n\n5. For $x \neq 0$, $$\frac{x}{2x} = \frac{1}{2}.$$\n\n6. Therefore, as $x$ approaches 0 from the left, the expression approaches $$\frac{1}{2}.$$\n\n7. Hence, the left-hand limit is $$\lim_{x \to 0^-} \frac{x}{x - |x|} = \frac{1}{2}.$$