Α7.ΟΡΙΟ ΣΤΟ ΑΠΕΙΡΟ

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    0x .

    . 1f (x)x 1

    =

    ( ) ( )A {1} ,1 1,= = + , f 0x 1= x 1 f(x) , , + . f + x 1

    1=+

    x 1lim

    . f:A IR , ( ) ( )0 0,x x , . :

    0 0 0x x x x x xlim f (x) lim f (x) lim f (x) +

    = + = = +

    0 0 0x x x x x xlim f (x) lim f (x) lim f (x) +

    = = =

    :

    0x xlim f (x) = + f(x)>0 0x

    0x x

    lim f (x) = f(x)

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    0x x

    lim f (x) = + 0x x1lim

    f(x)=0

    00

    x xx x

    0

    A lim f(x) 0 1 : limf(x) f(x) 0 x

    = = +>

    00

    x x

    x x0

    A lim f(x) 0 1 : limf(x) f(x) 0 x

    = = <

    0x x

    lim f (x) = + 0x x1lim 0

    f (x)=

    :

    0x xlim f (x) = + 0x xlim | f (x) | = +

    :

    0x xlim f (x) = + 0

    x xlim f(x)

    = + * : f(x) g(x) 0x

    0x xlim f (x) = + 0x xlim g(x) = +

    f (x) g(x) 0x 0x x

    lim g(x) = 0x xlim f (x) =

    . ( * )

    ) 2x 01limx

    = + 2x 01lim

    x= + ( )0 2x x 0

    1limx x

    = +

    ) x 0

    1limx+= + 2 1x 0

    1limx+ +

    = + ( )0 2x x 01lim

    x x+= +

    : , +

    ) x 0

    1limx= 2 1x 0

    1limx +

    = ( )0 2x x 01lim

    x x

    =

    : ,

    . : 1 ( )

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    x0 f : R R + + g : + + +

    gf + : + + ; ;

    2 ( )

    x 0

    f : >0 0

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