Dużym ułatwieniem w obliczaniu granic funkcji w punkcie
stanowią twierdzenia wynikające bezpośrednio z własności granic ciągów, a także
definicja Heinego granicy funkcji w punkcie.
Twierdzenie 1Jeżeli funkcja f jest określona w pewnym sąsiedztwie S(
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABYAAAATCAMAAACjpw26AAAAAXNSR0IArs4c6QAAAHVQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjpmOma2OpDbZgAAZgA6ZjoAZjo6ZmaQZpDbZrbbZrb/kDoAkDpmkGaQkNvbkNv/tmYAtmaQtv//25A625Bm27Zm29uQ29u22////7Zm/9uQ/9u2//+2///b9ZloTwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAkUlEQVQoU72QyxbCIAxEU2ylaq0vtCL4aBH+/xPNRHfGrVlAzpzJHQLRHysuRzUtLYKqT43qL+7A9mSbMfdzbvrK7DDvV0Tl9FgPR8Ylu6fbDNwolOLqM04MtNzRBx5ZoNwxcRL3W04bXGL0Igjkub23wyXkjtEGDPI87etQnGHA1VbIQAoe+FX6Ovryv75KoC/HBQnX7T9IzwAAAABJRU5ErkJggg==)
oraz:
a) funkcja f jest stała, tzn. f(x) = c, jeśli x
![](data:image/png;base64,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)
, to istnieje granica
funkcji f w punkcie
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAATCAMAAACuuX39AAAAAXNSR0IArs4c6QAAAGxQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6ZgAAZgA6ZjoAZjo6ZmaQZpDbZrbbZrb/kDoAkDpmkGaQkNvbkNv/tmYAtmaQtv//25A625Bm27Zm29uQ29u22////7Zm/9uQ/9u2//+2///bLbLkXQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAZUlEQVQoU82MSRKAIAwERxQR9wUVN0T4/x8Nlk/w4BxSnUpngL/ECm5ckQK2iFgN+OHIVa9pFw22mAC+S8Ywg5QRAQshnGyB8zFsyc17nAJd1Z6pWTtJFYw+pkT7jpG+iih0fZEbyy4FkvgAaaQAAAAASUVORK5CYII=)
i
![](data:image/png;base64,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)
;
b) f(x) = x, jeśli x
![](data:image/png;base64,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)
, to istnieje granica
funkcji f w punkcie
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAATCAMAAACuuX39AAAAAXNSR0IArs4c6QAAAGxQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6ZgAAZgA6ZjoAZjo6ZmaQZpDbZrbbZrb/kDoAkDpmkGaQkNvbkNv/tmYAtmaQtv//25A625Bm27Zm29uQ29u22////7Zm/9uQ/9u2//+2///bLbLkXQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAZUlEQVQoU82MSRKAIAwERxQR9wUVN0T4/x8Nlk/w4BxSnUpngL/ECm5ckQK2iFgN+OHIVa9pFw22mAC+S8Ywg5QRAQshnGyB8zFsyc17nAJd1Z6pWTtJFYw+pkT7jpG+iih0fZEbyy4FkvgAaaQAAAAASUVORK5CYII=)
i
![](data:image/png;base64,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)
.
Twierdzenie 2Jeżeli istnieją granice
![](data:image/png;base64,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)
oraz c jest dowolną liczbą rzeczywistą, to
istnieją również granice:
![](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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)
( przy dodatkowym założeniu, że
![](data:image/png;base64,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)
) i prawdziwe są 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,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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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)
=
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEIAAAAfCAMAAABkmuvzAAAAAXNSR0IArs4c6QAAAKhQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OgBmOjo6OjqQOmZmOmaQOma2OpDbZgAAZgA6ZgBmZjoAZjpmZjqQZmZmZpC2ZpDbZrbbZrb/kDoAkDo6kDpmkGY6kGZmkGaQkJA6kNvbkNv/tmYAtmY6tmZmtpCQtrbbtrb/ttuQttvbttv/tv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///bB3N4dwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAB3ElEQVRIS71ViVKDMBBNEIs3bdWqUE9o61FoLZT8/5+5R8IlOERn3JmkBcJj973NixD/HPvHNX9xezv8y5mUTmSWF9c78ze/ECKW8rC80Q+ZHST0MPNkINKgWrhA4NQCophCBeqe4ShS3w4i9yQkUUwRInWizAtENrKDECm+QBDFW3p28xnZQ8TIgi4kO0UObQspJqQK0rl6n01eXoVo0pl7mvYOYViRjJkHUVV4votBZhS1rgitMPI1cfhujFRA7J+arVWJSmR1B0Ko0K2pWV9XQQBZKgTVVOhEW89NNjD0yu7cyuYwrUVk4bT8CGd3xRiH6cOBEDmqRFMx9nmwAkgkciG7o0YnUUFTfpzQ0AoYiF6mSi6ob2jCWzRKfocVAlkv11D9ci1i2DswVOhvdCXDIEAHEgP2EbBITC4cYylNiLbl2Gx2ZqNlOXZ+wRB/shxCGGQ5q2+NrLnIL+WJ32s52oW42KvEuFKjwfOjSM2jPsvRLgROTTEyrtSAAKHh/X7LYRcqsxB0vZAudYbe7IF6wI7rthzjQgQBXPA17BjuUD5HttJ9hq5rWY45R7QLmTy0K4E/k0WzEvNI7GnjtS3HrOj6hSw2eFBwpJCKxQmo3wIuepzqpy//9tkXVOpK0zR/4+sAAAAASUVORK5CYII=)
=
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Twierdzenie 3Jeśli
![](data:image/png;base64,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)
=g, gdzie g > 0, to
![](data:image/png;base64,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)
.
Przykład 1
Oblicz:
oraz
=0
f(x) = ?
Podobnie jak ma to miejsce w przykładzie 1c w podrozdziale "Uzupełnienie wiedzy o granicach ciągu" należy pomnożyć
licznik oraz mianownik przez sprzężenie licznika, tak aby w liczniku otrzymać
wzór skróconego mnożenia (a-b)(a+b)=a^2-b^2, gdzie a=
oraz b=3:
= ![](data:image/png;base64,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)
Twierdzenie 4 (o trzech funkcjach)
Jeżeli funkcje f, g, h są określone w pewnym sąsiedztwie S(
) i dla dowolnej liczby x z
tego sąsiedztwa spełniona jest nierówność
oraz istnieją granice funkcji f i h w punkcie
i
=
=a, to istnieje granica
funkcji g w punkcie
i
=a.
Zadania do zrobienia
1. Oblicz granice:
a)
b)
c)
Odp. a)
b)
c)
2. Oblicz granice:
a)
b)
Odp. a)
b)