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L'Hôpital's Rule

Theorem

Let xx approach a∈R∪{−∞,+∞}a\in\mathbb{R}\cup\{-\infty,+\infty\} through an interval. Assume that, sufficiently near aa:

  • ff and gg are differentiable, and g′(x)≠0;g'(x)\neq0\text{;}
  • either f(x)→0f(x)\to0 and g(x)→0,g(x)\to0\text{,} or ∣f(x)∣→∞|f(x)|\to\infty and ∣g(x)∣→∞;|g(x)|\to\infty\text{;}
  • lim⁡x→af′(x)g′(x)=L,\displaystyle\lim_{x\to a}\frac{f'(x)}{g'(x)}=L\text{,} where LL may be finite or infinite.

Then

lim⁡x→af(x)g(x)=L.\lim_{x\to a}\frac{f(x)}{g(x)}=L.

Interpretation

L’Hôpital’s rule evaluates limits of quotients that have the indeterminate form 0/00/0 or ∞/∞\infty/\infty by comparing the rates of change of the numerator and denominator. It replaces the limit of f/gf/g by the limit of f′/g′f'/g', which is often simpler.

The curves e to the x minus 1 and sin 2x, both passing through the origin with tangent slopes 1 and 2; the ratio of the functions tends to the ratio of these slopes, 1/2.

lim⁡x→0f(x)g(x)=lim⁡x→0f′(x)g′(x)=f′(0)g′(0)=12\displaystyle\lim_{x\to0}\frac{f(x)}{g(x)}=\lim_{x\to0}\frac{f'(x)}{g'(x)}=\frac{f'(0)}{g'(0)}=\frac{1}{2}: near 00 both functions are close to their tangents (dashed), with slopes f′(0)=1f'(0)=1 and g′(0)=2g'(0)=2, so the quotient tends to the ratio of the slopes.

Zoom in towards the point: both graphs straighten into lines through (a,0)(a,0), and the quotient of their values approaches the quotient of their slopes.

Each hypothesis matters. The rule needs an indeterminate form: x+1x+2→12\frac{x+1}{x+2}\to\frac{1}{2} as x→0x\to0, while the quotient of the derivatives is 11. It also needs the limit of f′/g′f'/g' to exist: x+sin⁡xx→1\frac{x+\sin x}{x}\to1 as x→∞x\to\infty, but f′(x)g′(x)=1+cos⁡x\frac{f'(x)}{g'(x)}=1+\cos x has no limit. In such a case the rule gives no information, and the limit of f/gf/g must be found in another way.

The rule may be applied again as long as the form stays indeterminate: as x→∞x\to\infty, the quotient x2/exx^2/\mathrm{e}^x leads to 2x/ex2x/\mathrm{e}^x and then to 2/ex→02/\mathrm{e}^x\to0. Other indeterminate forms are first rewritten as a quotient of type 0/00/0 or ∞/∞\infty/\infty.

A common slip is to differentiate the quotient f/gf/g with the quotient rule. The rule differentiates numerator and denominator separately: it uses f′/g′f'/g', not (f/g)′(f/g)'.

Key Results

  • L’Hôpital’s rule applies only to indeterminate forms of type 0/00/0 or ∞/∞.\infty/\infty\text{.}
  • Differentiability of both numerator and denominator is required in a punctured neighbourhood of the point.
  • The rule must not be used to justify limits that are needed to establish the derivatives involved.
  • The rule needs f′(x)g′(x)\frac{f'(x)}{g'(x)} to have a limit L,L\text{,} finite or an infinite extended real number; if that limit does not exist, the rule gives no conclusion about f(x)g(x).\frac{f(x)}{g(x)}\text{.}

Examples

  • The limit lim⁡x→∞ln⁡xx=0\displaystyle \lim_{x\to\infty}\frac{\ln x}{x}=0 can be evaluated using L’Hôpital’s rule.

  • Although formally applicable, L’Hôpital’s rule cannot be used to justify

    lim⁡x→0sin⁡xx=1,\lim_{x\to 0}\frac{\sin x}{x}=1,

    because the derivative of sin⁡x\sin x depends on this limit itself. This would result in circular reasoning.

Prerequisites

Sources

Aliases: L'Hospital's rule, L'Hopital's rule, Bernoulli's rule Symbols: element of, infinity, real numbers, union

Last updated: 20 October 2026 03:02