1. **State the problem:** Find the limit of the function $$\frac{x^2 - 1}{x - 1}$$ as $x$ approaches 1. 2. **Recall the formula and rules:** Direct substitution gives $$\frac{1^2 - 1}{1 - 1} = \frac{0}{0}$$ which is an indeterminate form. We need to simplify the expression. 3. **Factor the numerator:** $$x^2 - 1 = (x - 1)(x + 1)$$. 4. **Simplify the expression:** $$\frac{(x - 1)(x + 1)}{x - 1} = x + 1$$ for $x \neq 1$. 5. **Evaluate the limit:** Now substitute $x = 1$ into the simplified expression: $$1 + 1 = 2$$. 6. **Conclusion:** The limit is $$2$$. Therefore, $$\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2$$.