1. The problem: Understand and apply L'Hôpital's Rule to evaluate limits that result in indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$.\n\n2. Statement of L'Hôpital's Rule: If $\lim_{x \to c} f(x) = 0$ and $\lim_{x \to c} g(x) = 0$ or both limits are $\pm \infty$, and the derivatives $f'(x)$ and $g'(x)$ exist near $c$, then\n$$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$$\nprovided the limit on the right side exists or is $\pm \infty$.\n\n3. Important rules: \n- The rule applies only to indeterminate forms $\frac{0}{0}$ or $\frac{\infty}{\infty}$.\n- You may need to apply the rule multiple times if the resulting limit is still indeterminate.\n- Always check the original limit to confirm it is an indeterminate form before applying the rule.\n\n4. Example: Evaluate $\lim_{x \to 0} \frac{\sin x}{x}$.\n- Direct substitution gives $\frac{0}{0}$, an indeterminate form.\n- Apply L'Hôpital's Rule: differentiate numerator and denominator:\n$$f'(x) = \cos x, \quad g'(x) = 1$$\n- New limit:\n$$\lim_{x \to 0} \frac{\cos x}{1} = \cos 0 = 1$$\n\n5. Explanation: L'Hôpital's Rule helps us find limits that are not directly solvable by substitution by using derivatives to simplify the ratio of functions.\n\nFinal answer: L'Hôpital's Rule states that for indeterminate forms $\frac{0}{0}$ or $\frac{\infty}{\infty}$, the limit of the ratio of functions equals the limit of the ratio of their derivatives, if that limit exists.