# [[LHopitalsRule]] (Mathematics > Calculus) ## Definition L'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms, such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$. If $f(x)$ and $g(x)$ are differentiable near $a$ and $ \lim_{x \to a} f(x) = 0 \quad \text{and} \quad \lim_{x \to a} g(x) = 0, $ or both approach $\pm\infty$, then $ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}, $ provided that the limit on the right-hand side exists. ## Key Concepts - **Indeterminate Forms:** Applies to limits of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$. - **Differentiability:** Both $f(x)$ and $g(x)$ must be differentiable near the point $a$. - **Simplification:** Converts a difficult limit into a simpler one using derivatives. - **Repeated Application:** May be used repeatedly if the result remains indeterminate. ## Important Properties 1. **Applicability:** Only valid when both $f(x)$ and $g(x)$ approach 0 or both approach $\infty$ as $x \to a$. 2. **Existence of Limit:** The limit $\lim_{x \to a} \frac{f'(x)}{g'(x)}$ must exist or be infinite. 3. **Consistency:** The rule can be applied multiple times if necessary. ## Essential Formulas - **L'Hôpital's Rule:** $ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}. $ ## Core Examples 1. **Basic Example:** Evaluate $\lim_{x \to 0} \frac{\sin x}{x}$. Direct substitution gives $\frac{0}{0}$, so applying L'Hôpital's Rule: $ \lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\cos x}{1} = \cos 0 = 1. $ 2. **Advanced Example:** Evaluate $\lim_{x \to 0} \frac{1-\cos x}{x^2}$. Substitution yields $\frac{0}{0}$; differentiate the numerator and denominator: $ \lim_{x \to 0} \frac{1-\cos x}{x^2} = \lim_{x \to 0} \frac{\sin x}{2x} = \lim_{x \to 0} \frac{\cos x}{2} = \frac{1}{2}. $ ## Related Theorems/Rules - **Squeeze Theorem:** Often used to evaluate limits when L'Hôpital's Rule applies. - **Taylor Series:** Provides an alternative method for approximating functions near a point. ## Common Pitfalls - **Improper Use:** Applying the rule when the limit is not in an indeterminate form. - **Neglecting Conditions:** Failing to ensure that both functions are differentiable near the point. - **Over-application:** Using L'Hôpital's Rule when simpler algebraic methods are available. ## Related Topics - [[Limit of Exponential Functions]] - [[Limits at Infinity for Exponential Functions]] - [[Limit of the Square Root Function]] - [[Limit of the Cube Root Function]] - [[Limits of Sine and Cosine]] - [[Limits of the Tangent Function]] - [[Limit of x over e^x as x approaches infinity]] - [[Limit of a Sequence]] - [[Continuity of a Function]] - [[Differentiability]] - [[Derivative and Tangent Line]] - [[Taylor Series]] - [[Taylor Polynomial]] - [[Linear Approximation]] - [[Horizontal Asymptotes of Rational Functions]] - [[Indeterminate Forms]] - **Indeterminate Forms:** Understanding forms like $\frac{0}{0}$ and $\frac{\infty}{\infty}$. - **Taylor Series:** For approximating functions and evaluating limits. - **Squeeze Theorem:** Another technique for evaluating limits. ## Quick Review Questions 1. Under what conditions can L'Hôpital's Rule be applied to a limit? 2. How would you use L'Hôpital's Rule to evaluate $\lim_{x \to 0} \frac{1-\cos x}{x^2}$? *** ![[LHopitalsRule_visualization]]