## You are here

Homel'H\^opital's rule

## Primary tabs

# l’Hôpital’s rule

L’Hôpital’s rule states that given an unresolvable limit of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, the ratio of functions $\frac{f(x)}{g(x)}$ will have the same limit at $c$ as the ratio $\frac{f^{{\prime}}(x)}{g^{{\prime}}(x)}$. In short, if the limit of a ratio of functions approaches an indeterminate form, then

$\lim_{{x\rightarrow c}}\frac{f(x)}{g(x)}=\lim_{{x\rightarrow c}}\frac{f^{{% \prime}}(x)}{g^{{\prime}}(x)}$ |

provided this last limit exists. L’Hôpital’s rule may be applied indefinitely as long as the conditions are satisfied. However it is important to note, that the nonexistance of $\lim\frac{f^{{\prime}}(x)}{g^{{\prime}}(x)}$ does not prove the nonexistance of $\lim\frac{f(x)}{g(x)}$.

Example: We try to determine the value of

$\lim_{{x\to\infty}}\frac{x^{2}}{e^{x}}.$ |

As $x$ approaches $\infty$ the expression becomes an indeterminate form $\frac{\infty}{\infty}$. By applying L’Hôpital’s rule twice we get

$\lim_{{x\to\infty}}\frac{x^{2}}{e^{x}}=\lim_{{x\to\infty}}\frac{2x}{e^{x}}=% \lim_{{x\to\infty}}\frac{2}{e^{x}}=0.$ |

Another example of the usage of L’Hôpital’s rule can be found here.

## Mathematics Subject Classification

26A24*no label found*26C15

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new question: Prove that for any sets A, B, and C, An(BUC)=(AnB)U(AnC) by St_Louis

Apr 20

new image: information-theoretic-distributed-measurement-dds.png by rspuzio

new image: information-theoretic-distributed-measurement-4.2 by rspuzio

new image: information-theoretic-distributed-measurement-4.1 by rspuzio

new image: information-theoretic-distributed-measurement-3.2 by rspuzio

new image: information-theoretic-distributed-measurement-3.1 by rspuzio

new image: information-theoretic-distributed-measurement-2.1 by rspuzio

Apr 19

new collection: On the Information-Theoretic Structure of Distributed Measurements by rspuzio

Apr 15

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

## Info

## Attached Articles

## Corrections

explanation by matte ✘

Accent by bbukh ✓

refer to example by archibal ✓

Accent by bbukh ✓