. x0 ‫ل‬ f ‫ی ت وا‬
‫ا رس‬

‫ا‬

‫بعت‬

‫ا‬

g

 g ( x ) = f ( x ) , x ≠ x0    g ( x0 ) = l 

(I
:
+∞ − ∞

∞×0

. lim f ( x ) = l x0 (7 f ( x) − l ≤ g ( x)

∞ ∞

0 : 0
+∞ + a = +∞ −∞ + a = −∞ +∞ + ∞ = +∞ −∞ − ∞ = −∞ +∞ − ∞

(1 (2

x0 x0 (a (b (c
(d

lim g ( x ) = 0

: a×∞ = ∞ ∞×∞ = ∞ 0×∞

lim g ( x ) = +∞ x0 x0 x0 f ( x) ≤ g ( x) lim f ( x ) = +∞

( a ≠ 0)

lim f ( x ) = −∞ x0 x0

f ( x) ≤ g ( x)

(a ∈ »)

lim g ( x ) = −∞ x0 lim f ( x ) = l x0 x0 x0 g ( x) ≤ f ( x) ≤ h ( x) lim g ( x ) = lim h ( x ) = l x0 .
. . :
+

(II f (a (1 (b (a (2 (b (3

∞ a =∞ =0 a ∞ ∞ 0 ∞ 0

a≠0 =∞ 0

. f ( ]a, b]) =  lim f , f ( b ) (* f ([ a, b ]) =  f ( a ) , f ( b ) (*    a    : . f ( ]a, b[ ) =  lim f , lim f  (*  b  a  
− +

:
.
∞

lim
∞

f

f ( x) ∞ = g ( x) ∞

(3
(a (b (* (*

f

([ a, b]) =  f ( b ) , f ( a ) (*   f lim ( f ( x ) + g ( x ) ) = +∞ − ∞ g ( x) f ( x)

( ∃c ∈ [ a, b]) : f ( c ) = λ

[ a, b ] f (b )

(a λ . g ( x) f ( x)

f (a)

. (b .
2   x = x ;x ≥ 0   x = − x2 ; x ≤ 0 

( ∃ c ∈ ]a , b [ ) : f ( c ) = 0
]a, b[

[ a, b ] f ( x) = 0

f

.

(a ≠ 0 ) (a ≠ 0 )

lim x0 f ( x) a − a 0 = = g ( x) 0 0 f ( x) 0 + 0 0 = = 0 0 g ( x)

(c (d

f ( a ) ⋅ f (b ) 〈 0

lim x0 . c ∈ [ a, b] . c f ( a ) ⋅ f (b ) ≤ 0 f (* : (* lim x →0

x2 = x : tan ( ax ) =1 ax

(e sin ( ax ) =1 ax

(III
:
I
J

1 − cos ( ax )

.

(4

(ax ) x0 2

=

1 2

lim x →0

lim x →0

(1 f . lim f ( x )

(5
(a

I

f

I

f f (I ) = J

(* (* (*

x0

f f lim f ( x ) = f ( x0 ) x0 .x

0

:

f −1 : J → I

f

(b

f

( ∀x ∈ J )( ∀y ∈ I ) : f −1 ( x ) = y ⇔ f ( y ) = x
J

.

f −1

.f
C f −1

J

f −1

Cf

..

(a (2 (b (c

(6 f lim f ( x ) = l ∈ » x0 x0

x0 lim f ( x ) x0

f

. (∆) : y = x

»

r'

r