proposition 7.18 Two-sided limit from one-sided limits

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:286 · p. 206

Rests on

depends_on definition 7.16 Limit ¶
- depends_on definition 7.2 Absolute value ¶
- depends_on definition 7.9 Real function ¶
  - depends_on definition 3.43 Map ¶
    - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.23 Predicate ¶
    - depends_on definition 3.28 Set ¶
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-6 ¶

Supports

depends_on lemma 7.33 Fermat: interior extremum ¶
- depends_on proposition 7.117 Lagrange multipliers in finitely many variables ¶
  - depends_on remark 7.118 The multiplier rule and its functional counterpart ¶
- depends_on proposition 16.16 Stationarity is necessary ¶
  - depends_on theorem 16.22 Euler–Lagrange ¶
    - depends_on definition 16.62 Field of extremals; slope function ¶
      - depends_on definition 23.52 Lagrangian family; caustic ¶
      - depends_on remark 23.8 The action as a function, not a functional ¶
      - depends_on theorem 16.63 Hilbert's invariant integral ¶
      - depends_on theorem 16.64 Weierstrass' sufficient condition ¶
    - depends_on phenomenon 30.52 Euler buckling ¶
      - depends_on example 30.53 A steel rod buckles at a twentieth of its crushing load ¶
      - depends_on remark 30.54 Other end conditions, and what buckling is variationally ¶
      - depends_on remark 30.55 Beyond the critical load: the elastica ¶
      - depends_on remark 30.68 Shells carry load in a different way ¶
    - depends_on proposition 29.39 Steady precession ¶
    - depends_on theorem 16.35 System of Euler–Lagrange equations ¶
      - depends_on proposition 16.6 The isoperimetric extremal is a circle ¶
      - depends_on theorem 16.41 Multiplier rule for pointwise constraints ¶
    - depends_on theorem 16.31 Natural boundary condition ¶
      - depends_on remark 30.67 Germain, Kirchhoff, and the edge conditions ¶
      - depends_on theorem A.691 Kirchhoff's edge conditions and the corner force ¶
      - depends_on theorem 16.32 Transversality ¶
  - depends_on theorem 16.51 Legendre's necessary condition ¶
    - depends_on corollary 16.65 Sufficiency for a weak minimum ¶
    - depends_on theorem 16.55 Jacobi's necessary condition ¶
      - depends_on proposition 23.53 A caustic point is a conjugate point ¶
      - depends_on remark 30.54 Other end conditions, and what buckling is variationally ¶ ↺
- depends_on theorem 7.34 Rolle ¶
  - depends_on theorem 7.37 Cauchy mean value theorem and l'Hôpital's rule ¶
    - depends_on lemma A.306 The standard smooth bump ¶
      - depends_on lemma A.307 Partition of unity on a compact manifold ¶
  - depends_on theorem 7.35 Mean value theorem ¶
    - depends_on corollary 7.36 cor:ana-mvt-consequences ¶
      - depends_on definition 7.41 Antiderivative ¶
      - depends_on lemma 7.67 The limit over real exponents ¶
      - depends_on lemma 7.75 The first quadrant ¶
      - depends_on lemma 7.78 Chord, arc, tangent ¶
      - depends_on lemma 9.10 Grönwall's inequality ¶
      - depends_on proposition 7.108 Euler's theorem on homogeneous functions ¶
      - depends_on proposition 7.61 The logarithm ¶
      - depends_on proposition 9.19 Separation of variables ¶
      - depends_on theorem 7.115 Constant rank ¶
    - depends_on lemma 7.93 A polynomial has at most $n$ roots ¶
    - depends_on lemma A.500 Graphs and $C^{1}$ images have zero content ¶
      - depends_on corollary A.514 Degeneracy on a negligible set ¶
      - depends_on lemma A.512 The boundary strip is thin ¶
    - depends_on lemma A.73 Differentiation under the integral sign ¶
      - depends_on lemma A.77 Poincaré lemma, converse form ¶
    - depends_on lemma A.287 The Newton map contracts ¶
      - depends_on lemma A.288 The iteration converges ¶
    - depends_on lemma A.290 $h$ is Lipschitz ¶
      - depends_on lemma A.291 $h$ is differentiable, with the stated derivative ¶
    - depends_on lemma A.440 Grönwall's inequality ¶
      - depends_on theorem A.441 Continuous dependence on the data and on the initial point ¶
    - depends_on lemma A.226 The level factor ¶
    - depends_on lemma A.468 The kernel is well defined, symmetric and Lipschitz ¶
      - depends_on theorem A.470 $T$ is compact ¶
    - depends_on lemma 11.68 Jensen's inequality for the logarithm ¶
    - depends_on proposition 7.105 Clairaut–Schwarz ¶
      - depends_on definition 10.4 The second-order operator ¶
      - depends_on lemma A.75 Differentiating a pullback along a flow ¶
      - depends_on lemma 22.20 The symplectic condition ¶
      - depends_on proposition 7.123 Second-order identities of the nabla calculus ¶
      - depends_on proposition 30.25 Twenty-one constants ¶
      - depends_on proposition 30.14 Saint-Venant compatibility is necessary ¶
      - depends_on proposition 22.30 Properties of the Poisson bracket ¶
      - depends_on proposition 13.148 prop:mfd-torsion-tensor ¶
      - depends_on proposition 10.16 Cauchy's characteristic strips ¶
      - depends_on theorem 7.132 Stokes ¶
      - depends_on theorem 7.106 Taylor's theorem in several variables ¶
      - depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
      - … 2 more
    - depends_on theorem 7.100 $C^{1}$ implies differentiable ¶
      - depends_on lemma A.291 $h$ is differentiable, with the stated derivative ¶ ↺
      - depends_on remark 7.101 Partial derivatives alone do not suffice ¶
      - depends_on theorem 7.112 Implicit function theorem ¶
      - depends_on theorem A.286 Implicit function theorem ¶
    - … 2 more
  - depends_on theorem 7.38 Taylor's theorem with Lagrange remainder ¶
    - depends_on definition 32.57 The Hénon–Heiles Hamiltonian ¶
    - depends_on definition 32.12 Linearization ¶
      - depends_on proposition 32.17 Classification of planar fixed points ¶
      - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
      - depends_on theorem 32.13 Linear stability ¶
    - depends_on lemma A.754 Exact depth expansion ¶
      - depends_on proposition A.755 The Boussinesq pair ¶
    - depends_on lemma A.187 Exponentiation ¶
    - depends_on lemma A.138 The flat exponential ¶
      - depends_on definition A.139 The standard radial mollifier ¶
      - depends_on lemma A.140 Properties of the mollifier ¶
    - depends_on lemma A.452 Young and Hölder ¶
      - depends_on lemma A.489 Courant–Lebesgue ¶
      - depends_on lemma A.463 Poincaré inequality; $B_{K}$ is an inner product ¶
      - depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶
    - depends_on lemma A.212 Second-order flatness ¶
      - depends_on lemma A.213 The three minima ¶
    - depends_on lemma A.209 Uniform third-order remainder ¶
      - depends_on corollary A.210 Uniform quadratic approximation on the $n^{-1/2}$ scale ¶
      - depends_on lemma A.211 Root-$n$ localisation ¶
    - depends_on phenomenon 28.13 Universality of small oscillations ¶
    - depends_on proposition 16.50 The second variation ¶
    - depends_on proposition 17.67 Holomorphy ¶
      - depends_on theorem 17.78 Causality implies dispersion relations ¶
      - depends_on theorem 17.93 Fundamental strip and holomorphy ¶
    - depends_on proposition 9.88 The model problem, and the error of the composite ¶
    - … 2 more
- depends_on theorem 16.104 Snell's law of refraction ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Limit	declared	parts/02-mathematical-methods/05-real-analysis.tex:294
`depends_on`	→	eq:ana-limit-left	declared	parts/02-mathematical-methods/05-real-analysis.tex:294
`depends_on`	→	eq:ana-limit-right	declared	parts/02-mathematical-methods/05-real-analysis.tex:294
`depends_on`	←	Fermat: interior extremum	declared	parts/02-mathematical-methods/05-real-analysis.tex:647
`proves`	←	ch:05-real-analysis@proof-6	declared	parts/02-mathematical-methods/05-real-analysis.tex:297