Math 2.0

Real functions Theorem

Exponential growth beats polynomial growth

Builds on Exponential functionLimit at infinity

Statement

For every natural number nn, lim⁡x→∞exxn=∞.\lim_{x \to \infty} \frac{e^x}{x^n} = \infty.

Remarks

For x>0x > 0, the series gives ex>xn+1(n+1)!e^x > \frac{x^{n+1}}{(n+1)!}. So exxn>x(n+1)!\frac{e^x}{x^n} > \frac{x}{(n+1)!}, which tends to infinity. Equivalently, xne−x→0x^n e^{-x} \to 0 as x→∞x \to \infty. The same holds for real exponents nn. So for each polynomial pp, p(x)e−x→0p(x) e^{-x} \to 0 as x→∞x \to \infty. Mathlib states the main claim as Real.tendsto_exp_div_pow_atTop.

Reading path

To understand this page from the ground up, read these 29 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionLimit at infinity, definition, no evidence levelLimit at infinitydefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionGeometric sequence, definition, no evidence levelGeometric sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionSum of a finite geometric series, statement, no evidence levelSum of a finite geome…statementMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionSqueeze theorem, statement, no evidence levelSqueeze theoremstatementPartial sum, definition, no evidence levelPartial sumdefinitionPower series, definition, no evidence levelPower seriesdefinitionThe sequence r^n tends to zero for |r| < 1, example, no evidence levelThe sequence r^n tend…exampleRadius of convergence, definition, no evidence levelRadius of convergencedefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionA series of nonnegative terms converges iff its partial sums are bounded, statement, no evidence levelA series of nonnegati…statementSum of a series, definition, no evidence levelSum of a seriesdefinitionAbsolute convergence, definition, no evidence levelAbsolute convergencedefinitionComparison test, statement, no evidence levelComparison teststatementGeometric series, example, no evidence levelGeometric seriesexampleRatio test, statement, no evidence levelRatio teststatementRatio formula for the radius of convergence, statement, no evidence levelRatio formula for the…statementThe exponential series has infinite radius of convergence, statement, no evidence levelThe exponential serie…statementExponential function, definition, no evidence levelExponential functiondefinitionExponential growth beats polynomial growth, statement, no evidence levelExponential growth be…statement
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. Sequence definition
  3. Limit of a function definition
  4. Real sequence definition
  5. Limit at infinity definition
  6. Bounded sequence definition
  7. Geometric sequence definition
  8. Limit of a sequence definition
  9. Monotone sequence definition
  10. Series definition
  11. Sum of a finite geometric series statement
  12. Monotone convergence theorem for sequences statement
  13. Convergent sequence definition
  14. Squeeze theorem statement
  15. Partial sum definition
  16. Power series definition
  17. The sequence r^n tends to zero for |r| < 1 example
  18. Radius of convergence definition
  19. Convergent series definition
  20. A series of nonnegative terms converges iff its partial sums are bounded statement
  21. Sum of a series definition
  22. Absolute convergence definition
  23. Comparison test statement
  24. Geometric series example
  25. Ratio test statement
  26. Ratio formula for the radius of convergence statement
  27. The exponential series has infinite radius of convergence statement
  28. Exponential function definition
  29. Exponential growth beats polynomial growth statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Exponential_growth, Other growth rates
About this page
Tag0CT6CBGTYY
URIm2://0CT6CBGTYY
Kindstatement, theorem
WritingW0 stub
MSC26A12, 33B10
StatusThis copy of the site has no build data. Run m2 status 0CT6CBGTYY.
UsesExponential function, Limit at infinity

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