1. Let's start by stating the problem: L'Hopital's Rule helps us find limits of indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$ when direct substitution in a limit gives these forms. 2. The rule states: 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 $$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$$ provided the limit on the right exists or is $\pm \infty$. 3. Important rules: - You can apply L'Hopital's Rule only when the original limit is an indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$. - Sometimes you may need to apply the rule multiple times. 4. Example: Find $\lim_{x \to 0} \frac{\sin x}{x}$. - Direct substitution gives $\frac{0}{0}$, an indeterminate form. - Apply L'Hopital's Rule: $$\lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\cos x}{1}$$ - Substitute $x=0$: $$\frac{\cos 0}{1} = \frac{1}{1} = 1$$ 5. So, the limit is 1. 6. Summary: L'Hopital's Rule transforms a difficult limit into a simpler one by differentiating numerator and denominator separately, making it easier to evaluate limits involving indeterminate forms.