1. The problem is to understand the limit notation $\lim_{x \to a}$ and what it represents in calculus. 2. The limit $\lim_{x \to a} f(x)$ describes the value that the function $f(x)$ approaches as $x$ gets arbitrarily close to $a$. 3. Important rules for limits include: - The limit may exist even if $f(a)$ is not defined. - The limit must approach the same value from both the left and right sides of $a$. 4. To evaluate a limit, substitute $x = a$ into $f(x)$ if possible. If direct substitution leads to an indeterminate form like $\frac{0}{0}$, use algebraic simplification, factoring, or special limit techniques. 5. Example: For $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$, direct substitution gives $\frac{0}{0}$. 6. Factor numerator: $x^2 - 4 = (x - 2)(x + 2)$. 7. Simplify: $\frac{(x - 2)(x + 2)}{x - 2} = x + 2$ for $x \neq 2$. 8. Now substitute $x = 2$: $2 + 2 = 4$. 9. Therefore, $\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4$. This process shows how limits help find values that functions approach near points where they might not be directly defined.