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L'Hôpital's rule for the $0 / 0$ form at finite or infinite, one-sided endpoints

Statement

Let $c \in R$ and let $f, g$ be differentiable on a deleted one-sided or two-sided neighbourhood of $c$ , with $g^{'} \neq 0$ there. Suppose $f (x) \to 0$ , $g (x) \to 0$ as $x \to c$ in the chosen mode. If $f^{'} (x) / g^{'} (x) \to L \in \overline{R}$ , then $f (x) / g (x) \to L$ in the same mode. The analogous statement at $+ \infty$ or $- \infty$ follows after the substitution $t = 1 / x$ , wherever the transformed functions are defined.

Facts & Assumptions

Given: The hypotheses and one fixed approach mode.

[L1]

Differentiability implies continuity, and the Cauchy quotient lemma gives a point between two arguments at which a secant quotient equals a derivative quotient (A function differentiable at $c$ is continuous at $c$ , Cauchy's mean-value theorem in quotient form when the denominator derivative is nonzero).

[L2]

Finite and infinite function limits have the quantified meanings in The $ε$ - $δ$ limit $\lim_{x \to c} f (x) = L$ of $f : A \to R$ 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 $\overline{R} = R \cup {- \infty, + \infty}$ , its order, and the arithmetic that is left undefined.

[L3]

Composition with $t = 1 / x$ is licensed by the chain rule, and ordinary finite limits obey their algebra laws (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients of function limits, the quotient under the hypothesis that the denominator limit is nonzero, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

Proof

technique · direct

1.1

Extend $f, g$ to $c$ by $f (c) = g (c) = 0$ . Their continuity at $c$ follows from the assumed zero limits, while differentiability gives continuity at every other point of the segment. For $x \neq c$ sufficiently close, the quotient lemma on the segment with endpoints $c, x$ gives $\frac{f (x)}{g (x)} = \frac{f^{'} (ξ_{x})}{g^{'} (ξ_{x})}$ , where $ξ_{x}$ lies strictly between $c$ and $x$ .

givenL1L2

2.1

As $x \to c$ in the chosen mode, $ξ_{x} \to c$ in that mode. Applying the defining finite or infinite limit inequality to the derivative quotient therefore gives $f (x) / g (x) \to L$ .

step 1.1L2

3.1

At infinity, put $F (t) = f (1 / t)$ , $G (t) = g (1 / t)$ . Then $F^{'} / G^{'} = f^{'} (1 / t) / g^{'} (1 / t)$ , since the common factor $- 1 / t^{2}$ cancels. Apply steps 1.1 and 2.1 as $t \to 0^{+}$ or $0^{-}$ , and translate back.

L3step 2.1algebra∎

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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 0/00/0 form at finite or infinite, one-sided endpoints