Funktsiooni piirväärtus

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Presentation transcript:

Funktsiooni piirväärtus Heldena Taperson www.welovemath.ee

Funktsiooni f(x) piirväärtuseks kohal a on arv A st kui argumendi x väärtused lähenevad kohale a, siis funktsiooni väärtused lähenevad arvule A kuitahes lähedale.

Piirväärtuse mõiste John Wallis 1655.a. Born November 23, 1616 Died October 28, 1703 Residence England Nationality English Fields Mathematics Institutions Queens' College, Cambridge University of Oxford Alma mater Emmanuel College, Cambridge Known for Wallis product Inventing the symbol Piirväärtuse mõiste John Wallis 1655.a.

Piirväärtuse teooria Augustin-Louis Cauchy Born21 August 1789 Paris, France Died23 May 1857 (aged 67) Sceaux, France Nationality French Fields Mathematics Institutions Ecole Centrale du Pantheon Ecole Nationale des Ponts et Chaussees Ecole Polytechnique Alma mater Ecole Nationale des Ponts et Chaussees Known for Cauchy integral theorem Cauchy-Schwarz inequality Augustin-Louis Cauchy

Karl Weierstrass Piirväärtuse teooria kaasaegne kuju Born October 31, 1815 Kingdom of Prussia Died February 19, 1897 (aged 81) Berlin, German Empire Residence Germany Nationality German Fields Mathematics Institutions Gewerbeinstitut Alma mater University of Bonn Münster Academy Known for Weierstrass function

Ühepoolsed piirväärtused

Lõpmatu piirväärtus

Piirväärtus lõpmatuse kohal Funktsiooni piirväärtus kohal lõpmatus on A , kui x väärtuse lõpmatul kasvamisel f(x) väärtused lähenevad arvule A.

Pea meeles! Määramatus on juhul, kui

JADA PIIRVÄÄRTUS Et jada võib vaadelda kui funktsiooni, mille määramispiirkonnaks on naturaalarvude hulk, siis jada piirväärtus on funktsiooni piirväärtuse, kus x läheneb lõpmatusele erijuhtum.

Jada, millel on lõplik piirväärtus, nimetatakse koonduvaks jadaks ja jada, millel ei ole lõplikku piirväärtust, nimetatakse hajuvaks jadaks. Jada piirväärtuse arvutamisel kasutatakse samu arvutusreegleid, mis lõpmatu piirväärtuse korral.