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:** The limit of a function as $x$ approaches a value can often be found by direct substitution. However, if direct substitution results in an indeterminate form like $\frac{0}{0}$, we need to simplify the expression. 3. **Check direct substitution:** Substitute $x = 1$ into the function: $$\frac{1^2 - 1}{1 - 1} = \frac{1 - 1}{0} = \frac{0}{0}$$ which is indeterminate. 4. **Simplify the expression:** Factor the numerator: $$x^2 - 1 = (x - 1)(x + 1)$$ So the function becomes: $$\frac{(x - 1)(x + 1)}{x - 1}$$ 5. **Cancel common factors:** For $x \neq 1$, cancel $x - 1$: $$x + 1$$ 6. **Evaluate the limit:** Now substitute $x = 1$ into the simplified function: $$1 + 1 = 2$$ **Final answer:** $$\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2$$