lemma 7.67 The limit over real exponents

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:1558 · p. 225

Rests on

depends_on corollary 7.36 cor:ana-mvt-consequences ¶
- depends_on theorem 7.35 Mean value theorem ¶
  - depends_on proposition 7.29 Linearity ¶
    - depends_on definition 7.11 Sum of functions ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on equation 7.14 eq:ana-derivh ¶
    - depends_on proposition 7.6 Algebra of limits ¶
      - depends_on definition 7.4 Convergence ¶
      - depends_on proposition 7.3 Triangle inequality ¶
      - proves proof ch:05-real-analysis@proof-3 ¶
    - proves proof ch:05-real-analysis@proof-13 ¶
  - depends_on theorem 7.34 Rolle ¶
    - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - proves proof ch:05-real-analysis@proof-17 ¶
    - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-9 ¶
    - proves proof ch:05-real-analysis@proof-18 ¶
  - proves proof ch:05-real-analysis@proof-19 ¶
- proves proof ch:05-real-analysis@proof-20 ¶
depends_on definition 7.19 Infinite limits ¶
- depends_on definition 7.2 Absolute value ¶
- depends_on definition 7.16 Limit ¶
  - depends_on definition 7.2 Absolute value ¶ ↺
  - depends_on definition 7.9 Real function ¶ ↺
depends_on lemma 7.65 The integer part ¶
- depends_on axiom 3.33 Peano axioms ¶
- depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
  - depends_on definition A.36 Cut ¶
  - depends_on theorem A.40 Least-upper-bound property ¶
    - depends_on definition A.36 Cut ¶ ↺
    - depends_on definition A.38 Order on $\R$ ¶
      - depends_on definition A.36 Cut ¶ ↺
    - proves proof app:A-long-proofs@proof-29 ¶
  - proves proof app:A-long-proofs@proof-32 ¶
- depends_on lemma 7.64 Well-ordering of $\N$ ¶
  - depends_on axiom 3.33 Peano axioms ¶ ↺
  - depends_on axiom 3.34 Principle of induction ¶
  - proves proof ch:05-real-analysis@proof-41 ¶
- proves proof ch:05-real-analysis@proof-42 ¶
depends_on proposition 7.57 The limit definition ¶
- depends_on definition 7.53 $\ee$ ¶
  - depends_on definition 7.45 Series ¶
    - depends_on definition 7.2 Absolute value ¶ ↺
    - depends_on definition 7.4 Convergence ¶ ↺
  - depends_on proposition 7.48 Ratio test ¶
    - depends_on proposition 7.47 Comparison; absolute convergence ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - depends_on definition 7.45 Series ¶ ↺
      - depends_on theorem 7.8 Cauchy criterion ¶
      - proves proof ch:05-real-analysis@proof-28 ¶
    - depends_on proposition 7.46 Geometric series ¶
      - depends_on corollary A.46 Monotone convergence ¶ ↺
      - depends_on definition 7.45 Series ¶ ↺
      - proves proof ch:05-real-analysis@proof-27 ¶
    - depends_on theorem 7.8 Cauchy criterion ¶ ↺
    - proves proof ch:05-real-analysis@proof-29 ¶
- depends_on equation 5.3 eq:lin-binomial ¶
- depends_on proposition 7.6 Algebra of limits ¶ ↺
- proves proof ch:05-real-analysis@proof-37 ¶
depends_on proposition 7.61 The logarithm ¶
- depends_on corollary 7.36 cor:ana-mvt-consequences ¶ ↺
- depends_on corollary A.46 Monotone convergence ¶ ↺
- depends_on equation 7.42 eq:ana-exp-derivative ¶
- depends_on lemma 7.60 Functional equation of the exponential ¶
  - depends_on corollary A.46 Monotone convergence ¶ ↺
  - depends_on definition 7.59 The real exponential ¶
    - depends_on definition 7.45 Series ¶ ↺
    - depends_on proposition 7.48 Ratio test ¶ ↺
    - depends_on theorem 7.50 Power series; radius of convergence ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
      - depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
      - depends_on proposition 7.46 Geometric series ¶ ↺
      - proves proof ch:05-real-analysis@proof-31 ¶
  - depends_on equation 5.3 eq:lin-binomial ¶ ↺
  - depends_on proposition 7.49 Cauchy product ¶
    - depends_on corollary A.46 Monotone convergence ¶ ↺
    - depends_on definition 7.45 Series ¶ ↺
    - depends_on proposition 7.6 Algebra of limits ¶ ↺
    - proves proof ch:05-real-analysis@proof-30 ¶
  - proves proof ch:05-real-analysis@proof-38 ¶
- depends_on proposition 7.32 Derivative of the inverse function ¶
  - depends_on definition 7.14 Inverse function ¶
    - depends_on definition 7.13 Composite function ¶
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on definition 3.47 Bijective map ¶
  - depends_on proposition 7.6 Algebra of limits ¶ ↺
  - depends_on theorem 7.23 Intermediate value theorem ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
    - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - proves proof ch:05-real-analysis@proof-8 ¶
  - proves proof ch:05-real-analysis@proof-16 ¶
- depends_on proposition 7.27 Differentiable implies continuous ¶
  - depends_on definition 7.20 Continuity at a point ¶ ↺
  - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
  - depends_on proposition 7.6 Algebra of limits ¶ ↺
  - proves proof ch:05-real-analysis@proof-11 ¶
- depends_on theorem 7.23 Intermediate value theorem ¶ ↺
- proves proof ch:05-real-analysis@proof-39 ¶
depends_on proposition 7.63 Laws of real powers ¶
- depends_on definition 7.62 Real powers ¶
  - depends_on proposition 7.61 The logarithm ¶ ↺
- depends_on equation 7.42 eq:ana-exp-derivative ¶ ↺
- depends_on lemma 7.60 Functional equation of the exponential ¶ ↺
- depends_on proposition 7.31 Chain rule ¶
  - depends_on definition 7.13 Composite function ¶ ↺
  - depends_on proposition 7.27 Differentiable implies continuous ¶ ↺
  - depends_on proposition 7.6 Algebra of limits ¶ ↺
  - proves proof ch:05-real-analysis@proof-15 ¶
- depends_on proposition 7.61 The logarithm ¶ ↺
- proves proof ch:05-real-analysis@proof-40 ¶
depends_on proposition 7.6 Algebra of limits ¶ ↺
proves proof ch:05-real-analysis@proof-43 ¶

Supports

depends_on proposition 7.68 The natural base ¶
- depends_on remark 7.70 What the condition says ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	cor:ana-mvt-consequences	declared	parts/02-mathematical-methods/05-real-analysis.tex:1563
`depends_on`	→	Infinite limits	declared	parts/02-mathematical-methods/05-real-analysis.tex:1563
`depends_on`	→	The integer part	declared	parts/02-mathematical-methods/05-real-analysis.tex:1563
`depends_on`	→	The limit definition	declared	parts/02-mathematical-methods/05-real-analysis.tex:1563
`depends_on`	→	The logarithm	declared	parts/02-mathematical-methods/05-real-analysis.tex:1563
`depends_on`	→	Laws of real powers	declared	parts/02-mathematical-methods/05-real-analysis.tex:1563
`depends_on`	→	Algebra of limits	declared	parts/02-mathematical-methods/05-real-analysis.tex:1563
`depends_on`	←	The natural base	declared	parts/02-mathematical-methods/05-real-analysis.tex:1628
`proves`	←	ch:05-real-analysis@proof-43	declared	parts/02-mathematical-methods/05-real-analysis.tex:1568