L’Hôpital’s rule

Definition

L’Hôpital’s Rule is a method for evaluating limits of indeterminate forms, typically $\frac{0}{0}$ or $\frac{\infty}{\infty}$. It involves taking the derivatives of the numerator and denominator until the limit can be evaluated.

5 Must Know Facts For Your Next Test

1. Only applies to indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$.
2. Requires differentiability of both the numerator and denominator at points near the point of evaluation.
3. The rule states that $\lim_{{x \to c}} \frac{f(x)}{g(x)} = \lim_{{x \to c}} \frac{f'(x)}{g'(x)}$ if the latter limit exists.
4. May need to apply L'Hôpital's Rule more than once if subsequent forms are still indeterminate.
5. Verify after applying that the resulting limit is not an indeterminate form before concluding.

Review Questions

• When can you apply L'Hôpital’s Rule?
• What should you do if applying L'Hôpital’s Rule once results in another indeterminate form?
• How do you verify that L'Hôpital’s Rule has been applied correctly?

Related terms

Indeterminate Forms: Expressions like $\frac{0}{0}$ or $\frac{\infty}{\infty}$ where limits cannot be directly computed without further analysis.

Derivative: The rate at which a function is changing at any given point, represented as $f'(x)$ for a function $f(x)$.

Limit:

The value that a function approaches as the input approaches some value, denoted as $\lim_{{x \to c}} f(x)$.