1. **State the problem:** We want to find the limit $$\lim_{x \to 1} \frac{x - 1}{x^2 - 1}$$. 2. **Recall the formula and rules:** The limit of a function as $x$ approaches a value is the value that the function approaches. If direct substitution leads to an indeterminate form like $\frac{0}{0}$, we simplify the expression. 3. **Evaluate the expression directly:** Substitute $x=1$: $$\frac{1 - 1}{1^2 - 1} = \frac{0}{0}$$ which is indeterminate. 4. **Simplify the denominator:** Note that $x^2 - 1$ is a difference of squares: $$x^2 - 1 = (x - 1)(x + 1)$$ 5. **Rewrite the limit:** $$\lim_{x \to 1} \frac{x - 1}{(x - 1)(x + 1)}$$ 6. **Cancel common factors:** For $x \neq 1$, cancel $x - 1$: $$\lim_{x \to 1} \frac{1}{x + 1}$$ 7. **Evaluate the simplified limit:** Substitute $x=1$: $$\frac{1}{1 + 1} = \frac{1}{2} = 0.5$$ 8. **Interpretation:** The values in the table approach 0.5 from both sides, confirming our limit. **Final answer:** $$\lim_{x \to 1} \frac{x - 1}{x^2 - 1} = 0.5$$