Definicja 1
Liczba g jest granicą ciągu nieskończonego (an) wtedy i
tylko wtedy, gdy dla każdej liczby dodatniej ε (epsilon) prawie wszystkie
wyrazy ciągu (an) znajdują się w odległości mniejszej niż ε od liczby g.
Jeśli liczba rzeczywista g jest granicą ciągu (an) to
zapisujemy:
lub
, jeśli
. Mówimy wówczas, że ciąg (an)
jest zbieżny do granicy g.
Prawie wszystkie wyrazy ciągu rozumiemy jako „wszystkie
wyrazy ciągu od pewnego miejsca”, więc:
Definicja 1.1
Liczba g jest granicą ciągu nieskończonego (an) wtedy i
tylko wtedy, gdy dla każdej liczby dodatniej ε istnieje taka liczba δ (delta),
że dla każdej liczby naturalnej n większej od δ zachodzi nierówność |an-g| <
ε.
Twierdzenie 1 (o działaniach arytmetycznych na granicach
ciągów zbieżnych)
Jeśli
i
, to istnieją granice ciągów
(an+bn),(an-bn),(an⋅bn) oraz
przy dodatkowym założeniu, że b≠0 i
≠0 dla każdej liczby
naturalnej dodatniej n, i prawdziwe są następujące równości:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAZCAMAAADKUMQKAAAAAXNSR0IArs4c6QAAAGlQTFRFAAAAAAAAAAA6ADpmADqQAGa2OgBmOjo6OjqQOmaQOma2OpC2ZgBmZjoAZpDbZrbbZrb/kDoAkDo6kGY6kNv/tmYAtmY6trbbttvbttv/tv//25A627Zm2////7Zm/9uQ/9u2//+2///bEFGa2AAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAYElEQVQYV4WOxw6AMAxDnVL2LFB2Gf3/j4RQwQ3hSyzZsR7wJduQkAZWpWb2gS0y0BWwSrMEQ48jF6Xyps/3JyCn395b0KK9/ZG58S3uKLnu7I3MYOuWGTjec+YJSRTACZ1ZBLlcXOanAAAAAElFTkSuQmCC)
Przykład 1
Oblicz:
a)
= ?
b)
= ?
c)
= ?
Ad. a)
=![](data:image/png;base64,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)
Ad. b) Granicę tego typu liczymy poprzez podzielenie
licznika oraz mianownika przez najwyższy wspólny czynnik, będący potęgą o
wykładniku (xn+m). W tym przypadku jest to
:
=![](data:image/png;base64,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)
Ad. c)
Metoda postępowania z tego
typu granicą jest niemal zawsze taka sama. Aby ją obliczyć należy pomnożyć
licznik i mianownik przez sprzężenie licznika tj. przez wyrażenie, generujące w
liczniku wzór skróconego mnożenia (a-b)(a+b)=a^2-b^2, gdzie a = n oraz b =
:
=![](data:image/png;base64,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)
Twierdzenie 2 (o trzech ciągach)
Jeśli dane są trzy ciągi nieskończone (an),(bn),(cn),
oraz istnieje taka liczba δ, że dla każdej
liczby naturalnej n większej od δ prawdziwa jest nierówność
, to
.
Przykład 2
Oblicz:
a)
= ?
b)
= ?
Ad. a) Z twierdzenia
o trzech ciągach wynika:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Ad. b) Z własności
funkcji cosx oraz na mocy twierdzenia o trzech ciągach:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGIAAAATCAMAAABcBShYAAAAAXNSR0IArs4c6QAAAJxQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOma2OpDbZgAAZgA6ZjoAZjo6ZmaQZma2ZpC2ZpDbZrbbZrb/kDoAkDpmkGY6kGaQkNvbkNv/tmYAtmY6tmaQtpBmttv/tv//25A625Bm27Zm27aQ29uQ29u229v/2////7Zm/9uQ/9u2//+2///bh1VklAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABK0lEQVRIS+2S2XLCMAxF5QCpWQoUaCG4bC0loUti0P//W2XhEGeBmcx0+gJ68cSR7pWODHCP2yVwmKz/cvgKuUg2djkL3PZqe2Y1JTmAKEhcC9x2velPvancmoKcFcosKLc5jTN9PRZiCECHNwPAhRDenP5q6cfH8cMpr1gDuY6LFqj6H27/WgYQzkkxgD1NGvoxvhFCXH6PVq8WbrEGrAUqcQrPQL80BSpulA/dXlNe2gCq1nlb9aawc6e7MLIUxyeCw23s5cBSDC2lMytnf1dBWUjp69AdhsFOG9+Io6LNmKsJfzrhvMJKC8ZVDqP9+UJT0CooJRmmFofnr/bqPf/Ss/KkJJf0aCePbJJfEUAohCETSWHQ08GfsGntUPHTqqhx5Cr7vl/+O4FfLsYgH3dGP1oAAAAASUVORK5CYII=)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFoAAAAaCAMAAAAALrtcAAAAAXNSR0IArs4c6QAAAJ9QTFRFAAAAAAAAAAA6ADo6ADqQAGa2OgAAOgBmOjoAOjo6OjqQOmZmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZmZmZma2ZpC2ZrbbZrb/kDoAkDo6kGY6kGaQkLb/kNvbkNv/tmYAtmZmtpBmtpCQtraQtrbbtrb/ttuQttv/tv//25A625Bm27Zm27aQ29v/2//b2////7Zm/9uQ/9u2//+2///bAIPwEwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABNUlEQVRIS+WVa1fCMAyG0zFREUXEuXrrQFEuXtph//9vM9uKx0Fa0iN+0X7ouu3Z2/TNTgLwF8bL4Xwfx9iWsePHfUjTMqVP2t4IkQPOyR2YrkjxcItEmW5OHpKS8UlbeaaXysoBmPT9WtlbDR9Pi5PsVf1Yuuxp1KhmcwDlxWl1B6Z+SI2YqE2qV5cYsV6d50sFBfrw8Dwa3k/Z0oUQYkDRVoq+rq2+AoRSXMu+LhLaD6/Md2k7Oaa/Jvfnw3ZylGRvzP88Aka0k/nStLldHCx7M2a8gL5TMOanGVVG3LKWbAfi3mxfmu0pOBhYhH21Oj8xTTi/84ewvY4DWznY9WkUvEvsv7wPVPe1BQykaRJtz0LV3ZEMxDWJzXT4q/sXyUCIJhGq7k6agVBNIlTdnTIDWTeJT2DUJEc+oqvCAAAAAElFTkSuQmCC)
oraz ![](data:image/png;base64,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)
= 0
Twierdzenie 3
Jeśli:
, to:
.
Twierdzenie 4
Jeśli:
, to: ![](data:image/png;base64,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)
Definicja 2
Ciąg nieskończony (an) nazywamy ciągiem rozbieżnym do plus
nieskończoności (odpowiednio ciągiem rozbieżnym do minus nieskończoności) wtedy
i tylko wtedy, gdy dla każdej liczby M istnieje taka liczba δ, że dla każdej
liczby naturalnej n > δ zachodzi nierówność an > M (odpowiednio an <
M).
Twierdzenie 5
Dane są nieskończone ciągi (an) i (bn) dla których
,
. Wówczas:
Jeśli b > 0, to ![](data:image/png;base64,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)
Jeśli b < 0, to ![](data:image/png;base64,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)
Przykład 3
Oblicz:
a)
= ?
b)
= ?
Ad. a)
=![](data:image/png;base64,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)
Ad. b)
= ![](data:image/png;base64,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)
Twierdzenie 6 Jeśli dany jest ciąg nieskończony (an), dla którego to .
Twierdzenie 6
Jeśli dany jest ciąg nieskończony (an), dla którego
to
.Twierdzenie 7
= ![](data:image/png;base64,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)
Twierdzenie 8
= ![](data:image/png;base64,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)
Zadania do zrobienia
1. Na podstawie twierdzenia o trzech
ciągach wyznacz granicę
Odp.
.
2. Czy prawdziwe jest zdanie: „Jeśli dany
jest ciąg nieskończony
(o
wyrazach niezerowych), dla którego
, to
lub
„?
Odpowiedź uzasadnij.
Odp. nie jest prawdziwe; np. jeśli
, gdzie
, to
i
nie istnieje.