# proof of De l’Hôpital’s rule

Let ${x}_{0}\in \mathbb{R}$, $I$ be an interval containing ${x}_{0}$ and let $f$ and $g$ be two differentiable functions defined on $I\setminus \{{x}_{0}\}$ with ${g}^{\prime}(x)\ne 0$ for all $x\in I$. Suppose that

$$\underset{x\to {x}_{0}}{lim}f(x)=0,\underset{x\to {x}_{0}}{lim}g(x)=0$$ |

and that

$$\underset{x\to {x}_{0}}{lim}\frac{{f}^{\prime}(x)}{{g}^{\prime}(x)}=m.$$ |

We want to prove that hence $g(x)\ne 0$ for all $x\in I\setminus \{{x}_{0}\}$ and

$$\underset{x\to {x}_{0}}{lim}\frac{f(x)}{g(x)}=m.$$ |

First of all (with little abuse of notation) we suppose that $f$ and $g$ are defined also in the point ${x}_{0}$ by $f({x}_{0})=0$ and $g({x}_{0})=0$. The resulting functions are continuous^{} in ${x}_{0}$ and hence in the whole interval $I$.

Let us first prove that $g(x)\ne 0$ for all $x\in I\setminus \{{x}_{0}\}$. If by contradiction^{} $g(\overline{x})=0$ since we also have $g({x}_{0})=0$, by Rolle’s Theorem we get that ${g}^{\prime}(\xi )=0$ for some $\xi \in ({x}_{0},\overline{x})$ which is against our hypotheses.

Consider now any sequence ${x}_{n}\to {x}_{0}$ with ${x}_{n}\in I\setminus \{{x}_{0}\}$. By Cauchy’s mean value Theorem there exists a sequence ${x}_{n}^{\prime}$ such that

$$\frac{f({x}_{n})}{g({x}_{n})}=\frac{f({x}_{n})-f({x}_{0})}{g({x}_{n})-g({x}_{0})}=\frac{{f}^{\prime}({x}_{n}^{\prime})}{{g}^{\prime}({x}_{n}^{\prime})}.$$ |

But as ${x}_{n}\to {x}_{0}$ and since ${x}_{n}^{\prime}\in ({x}_{0},{x}_{n})$ we get that ${x}_{n}^{\prime}\to {x}_{0}$ and hence

$$\underset{n\to \mathrm{\infty}}{lim}\frac{f({x}_{n})}{g({x}_{n})}=\underset{n\to \mathrm{\infty}}{lim}\frac{{f}^{\prime}({x}_{n})}{{g}^{\prime}({x}_{n})}=\underset{x\to {x}_{0}}{lim}\frac{{f}^{\prime}(x)}{{g}^{\prime}(x)}=m.$$ |

Since this is true for any given sequence ${x}_{n}\to {x}_{0}$ we conclude that

$$\underset{x\to {x}_{0}}{lim}\frac{f(x)}{g(x)}=m.$$ |

Title | proof of De l’Hôpital’s rule |
---|---|

Canonical name | ProofOfDeLHopitalsRule |

Date of creation | 2013-03-22 13:23:31 |

Last modified on | 2013-03-22 13:23:31 |

Owner | paolini (1187) |

Last modified by | paolini (1187) |

Numerical id | 10 |

Author | paolini (1187) |

Entry type | Proof |

Classification | msc 26A24 |

Classification | msc 26C15 |