1. The problem is to understand how to solve a limit in calculus. 2. The limit of a function $f(x)$ as $x$ approaches a value $a$ is written as $$\lim_{x \to a} f(x)$$ and it represents the value that $f(x)$ approaches as $x$ gets closer to $a$. 3. Important rules include: - If direct substitution of $a$ into $f(x)$ gives a finite number, that is the limit. - If substitution gives an indeterminate form like $\frac{0}{0}$, use algebraic simplification, factoring, rationalizing, or L'Hôpital's Rule. 4. Example: Solve $$\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$$ 5. Direct substitution gives $$\frac{2^2 - 4}{2 - 2} = \frac{0}{0}$$ which is indeterminate. 6. Factor numerator: $$x^2 - 4 = (x - 2)(x + 2)$$ 7. Rewrite limit: $$\lim_{x \to 2} \frac{(x - 2)(x + 2)}{x - 2}$$ 8. Cancel common factor: $$\lim_{x \to 2} \frac{\cancel{(x - 2)}(x + 2)}{\cancel{(x - 2)}} = \lim_{x \to 2} (x + 2)$$ 9. Now substitute $x = 2$: $$2 + 2 = 4$$ 10. So, the limit is 4. This method applies to many limits where direct substitution leads to indeterminate forms.