Gränsvärden
Introduktionsföreläsning i två grupper
Vad händer med uttrycket när x närmar sig 4?
Lösning i WolframAlpha: http://www.wolframalpha.com/input/?i=lim+x-%3E4+f%28x%29+%3D+%28x%5E%280.5%29+-+2%29+%2F+%28x²+-+5x+%2B+4%29
Lösning i tabell-GGB
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Grafen
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Här kommer text om gränsvärden.
Upplägget.
Motivering.
Omgivningar.
Intervall
Om vi tänker oss alla tal mellan två tal a och b så kallas det ett intervall. Det finns intervall av tre typer. Öppna intervall, slutna intervall och halvöppna intervall (se figurer). Här gör man så att man ritar öppna cirklar när punkten inte ingår i intervallet och fyllda cirklar när det ingår. Intervallet [math]\displaystyle{ ]1 ,4[ }[/math] är alltså ett öppet intervall dvs 1<x<4. Det indikeras av att hakparenteserna inte sluter om.
På samma sätt är [math]\displaystyle{ [0.5 , 5] }[/math] slutet.[math]\displaystyle{ ( 0.5\le x\le5 ) }[/math] och de två sista intervallen halvöppna.
Alltså
| Definition |
|---|
| Ett öppet intervall ]a,b[ består av alla tal x mellan a och b utom a och b ; a<x<b |
| Uppgift
|
Rita tallinjer och lägg in intervallen 2<x≤3 ; 4<x<6 ; 1≤x≤1.1
Du kan rita i Geogebra. Du kan också rita på eget papper eller trycka ut detta papper |
Facit: (klicka expandera till höger)
| Uppgift |
|---|
lägg också in intervallet nedan på en ytterligare tallinje
|
Oändlikhetsymbolen [math]\displaystyle{ \infty }[/math] kommer att förklaras mer senare.
Inre punkt i ett intervall
Om en punkt A finns inne i ett intervall kallas den inre punkt i till intervallet.
plats för figur
| Definition |
|---|
| En punkt A som ligger ligger helt inne i ett intervall kallas inre punkt till intervallet.
|
| Uppgift |
|---|
Vilket eller vilka av talen [math]\displaystyle{ 1 ; 1.414 ; \sqrt{2} ; 3 ; \pi }[/math] är inre punkter till intervallen
|
Facit: (klicka expandera till höger)
- 1 [math]\displaystyle{ \sqrt{2}, 3 }[/math] därför att 1.414 ingår inte (öppet) och [math]\displaystyle{ \pi }[/math] är inte inre punkt! det är endast en (kant)punkt ett slutet intervall
- 2 3 och [math]\displaystyle{ \pi }[/math]
Omgivning
| Definition |
|---|
| Om en punkt A är inre punkt till ett öppet intervall U kallas U en omgivning till A |
Ofta kommer vi att använda symmetriska omgivningar till en punkt som [math]\displaystyle{ A-\epsilon\lt A\lt A+\epsilon }[/math]
där [math]\displaystyle{ \epsilon }[/math] är ett godtyckligt positivt tal > 0 (ofta litet) tal. Det kan också skrivas [math]\displaystyle{ ]A-\epsilon, A+\epsilon[ }[/math].
| Uppgift |
|---|
| Uppgifter på omgivningar |
Punkterade omgivningar
Ibland undantar man A från själva omgivningen till A då kallas det en punkterad.
| Definition |
|---|
| De sammanslagna intervallen [math]\displaystyle{ P_-= \rm{A-a\lt x\lt A} }[/math] och [math]\displaystyle{ P_+=\rm{A\lt x\lt A+b} }[/math] kallas en punkterad omgivning P till A
Det kan också skrivas så här: P är alla x som uppfyller [math]\displaystyle{ ]a,A[ och ]A,b[ }[/math] där a<A och b>A |
plats för figur
| Uppgift |
|---|
| uppgifter punkterade omgivningar |
Vänster och höger omgivningar
Oegentliga gränsvärden
Gränsvärden.
Alternativa definitioner.
Facit till vissa uppgifter
GeoGebra
Tangent och sekant
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Övning gränsvärden
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Arean mellan kurvorna
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