Math 2.0

Real functions Differentiation Theorem

L'Hôpital's rule

Builds on Cauchy mean value theoremLimit of a functionIndeterminate form

Statement

Let II be an open interval and let c∈Ic \in I. Let f,gf, g be real functions differentiable on I∖{c}I \setminus \{c\}, with g′(x)≠0g'(x) \ne 0 for all x∈I∖{c}x \in I \setminus \{c\}. Assume that f(x)→0f(x) \to 0 and g(x)→0g(x) \to 0 as x→cx \to c, and that f′(x)/g′(x)→Lf'(x)/g'(x) \to L as x→cx \to c. Then f(x)/g(x)→Lf(x)/g(x) \to L as x→cx \to c.

Remarks

The proof uses the Cauchy mean value theorem on intervals between cc and xx. Similar forms hold for the indeterminate form ∞/∞\infty/\infty, for one-sided limits, and for x→±∞x \to \pm\infty. For example, lim⁡x→01−cos⁡xx2=lim⁡x→0sin⁡x2x=12\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \lim_{x \to 0} \frac{\sin x}{2x} = \frac{1}{2}. Mathlib states a form as deriv.lhopital_zero_nhds.

History

Johann Bernoulli found the rule and sent it to Guillaume de l'Hôpital in a letter of 1694. L'Hôpital published it in his textbook of 1696.

Reading path

To understand this page from the ground up, read these 25 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLocal maximum, definition, no evidence levelLocal maximumdefinitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionLocal minimum, definition, no evidence levelLocal minimumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionIndeterminate form, concept, no evidence levelIndeterminate formconceptBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionFermat's theorem on stationary points, statement, no evidence levelFermat's theorem on s…statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementBoundedness theorem, statement, no evidence levelBoundedness theoremstatementExtreme value theorem, statement, no evidence levelExtreme value theoremstatementRolle's theorem, statement, no evidence levelRolle's theoremstatementMean value theorem, statement, no evidence levelMean value theoremstatementCauchy mean value theorem, statement, no evidence levelCauchy mean value the…statementL'Hôpital's rule, statement, no evidence levelL'Hôpital's rulestatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Limit point of a set definition
  2. Local maximum definition
  3. Sequence definition
  4. Limit of a function definition
  5. Local minimum definition
  6. Real sequence definition
  7. Subsequence definition
  8. Continuity at a point definition
  9. Derivative definition
  10. Indeterminate form concept
  11. Bounded sequence definition
  12. Limit of a sequence definition
  13. Monotone sequence definition
  14. Continuous function definition
  15. Fermat's theorem on stationary points statement
  16. Convergent sequence definition
  17. Every sequence has a monotone subsequence statement
  18. Monotone convergence theorem for sequences statement
  19. Bolzano–Weierstrass theorem statement
  20. Boundedness theorem statement
  21. Extreme value theorem statement
  22. Rolle's theorem statement
  23. Mean value theorem statement
  24. Cauchy mean value theorem statement
  25. L'Hôpital's rule statement

Further reading

  1. Web https://en.wikipedia.org/wiki/L%27Hôpital%27s_rule, General form
About this page
Tag0JWT4PPQ7B
URIm2://0JWT4PPQ7B
Aliaseslhopital-rule, bernoulli-rule
Kindstatement, theorem
WritingW1 stated
MSC26A24
StatusThis copy of the site has no build data. Run m2 status 0JWT4PPQ7B.
UsesCauchy mean value theorem, Limit of a function, Indeterminate form
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