Limite
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