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L'Hôpital's rule for the $\infty/\infty$ form at finite or infinite, one-sided endpoints

Statement

Let $f,g$ be differentiable on a one-sided neighbourhood of $c$ , or on a tail at $+\infty$ or $-\infty$ , with $g'\ne0$ . Suppose $|f(x)|\to\infty$ and $|g(x)|\to\infty$ in the selected mode, with each numerator and denominator eventually of fixed sign. If $f'(x)/g'(x)\to L\in\overline{\mathbb R}$ , then $f(x)/g(x)\to L$ .

Facts & Assumptions

Given: The stated hypotheses in one fixed approach mode.

[L1]

Differentiability implies continuity, and the Cauchy quotient lemma compares increments of $f$ and $g$ with a derivative quotient at an intermediate point (A function differentiable at $c$ is continuous at $c$ , Cauchy's mean-value theorem in quotient form when the denominator derivative is nonzero).

[L2]

The relevant finite and infinite limits are exactly those of The $\varepsilon$ - $\delta$ limit $\lim_{x \to c} f(x) = L$ of $f : A \to \mathbb{R}$ at a limit point $c$ of $A$ , The left and right limits of $f$ at $c$ , as limits of the restrictions of $f$ to $A \cap (-\infty, c)$ and $A \cap (c, \infty)$ , Limits at $+\infty$ and $-\infty$ , and infinite limits at a point, and The extended real line $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$ , its order, and the arithmetic that is left undefined.

Proof

technique · direct

1.1

Fix a base point $a$ inside the domain. For variable $x$ farther toward the limiting end, [L1] gives $\frac{f(x)-f(a)}{g(x)-g(a)}=\frac{f'(\xi_x)}{g'(\xi_x)}$ , where $\xi_x$ lies between $a$ and $x$ .

givenL1

2.1

First choose $a$ sufficiently far toward the end that the derivative quotient is as close to $L$ as required throughout the remaining tail. Then the quotient of increments has the same bound for every later $x$ .

step 1.1L2choose

3.1

Since $|g(x)|\to\infty$ , $g(a)/g(x)\to0$ ; since the increment quotient is bounded in the finite- $L$ case, the identity $\frac{f(x)}{g(x)}=\frac{f(x)-f(a)}{g(x)-g(a)}\left(1-\frac{g(a)}{g(x)}\right)+\frac{f(a)}{g(x)}$ gives the finite conclusion. For $L=\pm\infty$ , choose the derivative-quotient lower or upper bound first and then make the two fixed-base terms negligible, obtaining the defining arbitrary bound.

step 2.1L2algebra

4.1

Thus the quotient has limit $L$ in every stated mode.

step 3.1∎

Depends on

Used by

Scope, endpoint, factorial, and deferred-remainder conventions Remark

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 63 results over 20 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Lebl, Basic Analysis I, Taylor's theorem and related calculus (standard reference, not scraped)
MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 lecture notes (standard reference, not scraped)
UC Davis, L'Hopital's rule (standard reference, not scraped)
Colgate University MATH 323, Chapter 5 notes (standard reference, not scraped)

L'Hôpital's rule for the ∞/∞\infty/\infty∞/∞ form at finite or infinite, one-sided endpoints