'ا (ز
ر ا% & ا !"# ا$ : إز ا ذ ى او م ر ا
[email protected] !) )رات ا+ ا . ر-. / ر و ا )وال ا- )ود و ا )وال ا$ ب ت ا )وال ا3 ــ . د56 (ل ا )وال ا7 85 9 ا5:: ب ت ا )وال ا3 ــ
"5;" ر x + 3x = 0 أن59 2
. ]-1,1[ , 0 B'ي "آ- اE+ ح اG "ا ل ا9( .
K و
x 2 +3 x < 4 x
x+3 < 4
إذن
نJ!
2< x+3< 4
. 2 5L% اM 3 lim x x →0
2
−1 < x < 1
lim x x →0
أن7 و
+ 3x = 0 نJ!
2
+ 3x = 0
: 2 =>" ر .
x +1 x −1
−3 =
x + 1 − 3x + 3
=
x −1
−2 x + 4 x −1
x +1 f ( x) = x −1
(
−2 x − 2
=
)
x −1
إذن
2
x −2 < x−2 x −1
1
و ن
4 lim =0 x→+∞ x 2 + 5
4
x 2 ن+
2
x +5
4 lim =0 x→+∞ x 2
=
4 2
x +5
<
: ' أن 4 x2
: )*
2 ,-. ا/01 %& و
4 lim =0 x→+∞ x 2 + 5
x = 0 أنO: 7 57 x →+∞ x + 1 lim
2
: 4 =>" ر
2x −1 lim f ( x) = 2 ' أن x →+∞ x2 + 4 2 2 2 2x −1 2x −1 − 2x − 8 −9 9 f ( x) − 2 = 2 −2 = = 2 = 2 )* 2 x +4 x +4 x +4 x +4 f ( x) =
2
2x2 −1 9 lim =2 2) ا/01 إذنlim 2 = 0 و x →+∞ x 2 + 4 x →+∞ x
[email protected]
9 9 < 2 x +4 x 2
)* و
2x2 −1 lim =2 x →+∞ x 2 + 4 3x 2 − 1 lim 2 = 3 أنO: 7 57 x →+∞ x + 2
: 5 =>" ر x +1 f ( x) = lim f ( x) = −1 ' أن x →−∞ −x lim f (− x) = −1 ف ' أن45 2) ا67
x →+∞
lim f (− x) = −1 نlim
x →+∞
x →+∞
1 = 0 و ان x
f (− x) − ( −1) =
−x +1 −x +1+ x 1 1 +1 = = = )* x x x x . lim f ( x) = −1 و x →−∞
−x +1 lim = −1 x →−∞ x
: 6 =>" ر
1+ x = +∞ ' أن x2 2
. "1 *د9 :; . lim x →0
lim x →0
1 + x2 1 ≥ 2 : )* 2 x x
1 + x 2 ≥ 1 ن+
1 + x2 1 = +∞ إذنlim 2 = +∞ )* و 2 x →0 x x
1 + x2 = +∞ x2
lim x →0
x +1 = +∞ أنO: 7 57 x4 2
lim x →0
: 7 =>" ر . lim x →1
f (1 + h) =
3 > أن0 و h2
3
( x − 1)
f ( x) =
lim f ( x) = lim f (1 + h) = lim x →1
lim x →1
3
( x − 1)
2
h →0
2
h →0
= +∞ ' أن 3
( x − 1)
1 = +∞ h2
2
" ر . lim x + 2 x = +∞ ' أن 4
x + 2 x = x ( x + 2) > x 4
3
x →+∞
وx + 2 > 1 " وx > 1 ض أنB +∞ ار4 ,)@ و أن اx 4 + 2 x = x ( x3 + 2 ) : )* 3
. lim x 4 + 2 x = +∞ إذنlim x = +∞ أن%&و x →+∞
lim x + 2 x = +∞ 4
x →+∞
[email protected]
x →+∞
Y( ا+"87 ا ت : )ود$ ا )وال ا : )ا7 KرL ه+5+3 ــ ا ! )د1
lim 3 x 2 − 2 x + 4 x →0
lim 3x − 2 x + 4 = 3 × 02 − 2 × 0 + 4 = 4 2
x →0
lim 4 x 2 + x − 1
x →−2
lim 4 x 2 + x − 1 = 4 × ( −2 ) + ( −2 ) − 1 = 16 − 2 − 1 = 13 2
x →−2
: "أس9 أآK) ي- ) ا$ ه ا−∞ أو+∞ ! ــ ا1 lim 2 x − x − 5 2
x →+∞
lim 2 x − x − 5 = lim 2 x 2 = +∞ 2
x →+∞
x →+∞
lim 3x − 2 x 2 − 5 x 5
x →−∞
lim 3x5 − 2 x 2 − 5 x = lim 3 x5 = −∞
x →−∞
x →−∞
lim − 2 x + 3x − 2 4
x →+∞
lim − 2 x 4 + 3 x − 2 = lim − 2 x 4 = −∞
x →+∞
x →+∞
: ر- ا )وال ا : ـــ ا ()د7 (ض ا ل+5+3 ر ! )د-. )) دا$ ــ1 2x − 3 lim x →−1 x2 2 x − 3 2 ( −1) − 3 −5 lim = = = −5 2 x →−1 x2 1 ( −1) lim x→2
lim x→2
−2 x + 3
( x − 2)
2
−2 x + 3
( x − 2)
=
2
−4 + 3 −1 = =0 02 +∞
3x + 1 x →1 x − 1 3x + 1 4 lim− = − = −∞ x →1 x − 1 0 lim−
: )د$ "5] O^_ 58( ^ أنY( ــ ا2
x −1 x −1 x2 −1 0 )د$ "5] O^_ lim = x →1 x − 1 0 2 ( x − 1)( x + 1) = lim x + 1 = 2 x −1 lim = lim x →1 x − 1 x →1 x →1 x −1 : "أس9) اآ$ رج اa ه−∞ أو+∞ ! ــ ا3 2 3x − 2 x lim x →+∞ 2 x 2 + 5 3x 2 − 2 x 3x 2 3 3 lim = lim = lim = 2 2 x →+∞ 2 x + 5 x →+∞ 2 x x →+∞ 2 2 5 2 3x − 2 x lim x →−∞ 2 x 2 + 5 x
[email protected] 2
lim x →1
3x5 − 2 x 2 3x5 3x3 lim = lim 2 = lim = −∞ x →−∞ 2 x 2 + 5 x x →−∞ 2 x x →−∞ 2
: ر-. / ا )وال ا : ـــ ا ()د7 (ض ا ل+5+3 ر ! )د-. )) دا$ ــ1
lim x→2
x2 −1 x +1
x2 − 1 3 1 = = x +1 3 3
lim x→2
x2 − 1 x→2 x−2 2 x −1 3 lim+ = + = +∞ x→2 x−2 0 lim+
x2 − 4 x−2
lim+
x→2
x2 − 4 0 )د$ "5] O^_ lim = x → 2+ x−2 0 2 2 x + 2) x − 4 x + 2) x2 − 4 ( ( ( x + 2 ) = 4 = +∞ x −4 lim+ = lim+ = lim+ = lim + x→2 x → 2 ( x + 2 )( x − 2 ) x →2 x → 2+ x−2 x2 − 4 x2 − 4 0 : ر-. )ا−∞ أو+∞ ! ــ ا2
lim
x →+∞
x 2 + 2 = +∞ = +∞
lim
x →+∞
lim
x →+∞
)د$ "5] O^_ lim
x →+∞
x +2 2
x2 + 2 − 2x
x 2 + 2 − 2 x = +∞ − ∞
lim
x →+∞
x 2 + 2 − 2 x = lim x 1 + x →+∞
2 2 − 2 x = lim x 1 + 2 − 2 2 x →+∞ x x
x2 + 2 = x 1 +
Y( ا O5( ا+";
2 نC x2
2 − 2 = 1 + 0 + − 2 = 1 − 2 = −1 و 2 x 2 x 2 + 2 − 2 x = −∞ إذنlim x 1 + 2 − 2 = −∞ و x →+∞ x
lim x = +∞ وlim 1 +
x →+∞
lim
x →+∞
x →+∞
lim
x →+∞
x 2 + 2 +∞ = x →+∞ x +∞ 2 x 1+ 2 2 x +2 x = lim 1 + 2 = 1 + 0 + = 1 = lim x →+∞ x →+∞ x x x2
)د$ "5] O^_
lim
x →+∞
x2 + 2 x
lim
lim
x →+∞
x2 + 1 − x
[email protected]
Y( ا
O5( ا
)د$ "5] O^_ )د$ "5] O^_
lim
x →+∞
x 2 + 1 − x = lim x 1 + x →+∞
x→+∞
O5( ا
1 =0 +∞
P!ا "ا
2
2
2
x→+∞
x2+1+x
Y( ا
1 1 − x = lim x 1 + − 1 = +∞ × 0 x →+∞ x2 x2
x +1−x)( x +1+x) ( x +1−x = lim = lim x +1−x 2
lim
x 2 + 1 − x = +∞ − ∞
lim
x →+∞
2
2 x→+∞ x +1+x
= lim
+1
2 x→+∞ x +1+x
=
1 − cos x 1 tan x sin x = وlim = 1 وlim =1 2 x → 0 x → 0 x 2 x x sin x lim x →0 4 x sin x 1 sin x 1 1 lim = lim = ×1 = x →0 4 x 4 x →0 x 4 4 sin x lim x → 0 sin 5 x sin x sin x x 5x 1 1 lim = lim × × = 1× × 1 = x → 0 sin 5 x x →0 x 5 x sin 5 x 5 5 1 − cos 2 x lim x →0 x2 1 − cos 2 x 1 − cos 2 x 1 − cos 2 x 1 lim = lim 4 × = 4 × lim = 4× = 2 2 2 2 x →0 x →0 x →0 x 2 ( 2x) ( 2x ) lim x →0
2 x4 x → 0 cos x 2 − 1
lim
−2 ( x 2 ) x2 ) ( 2 x4 lim = lim = −2 lim = −2 × 2 = −4 x → 0 cos x 2 − 1 x → 0 1 − cos x 2 x → 0 1 − cos x 2 2
2
sin 2 x x →0 x sin 2 x sin 2 x sin 2 x lim = lim 2 × = 2 × lim = 2 ×1 = 2 x →0 x → 0 x → 0 x 2x 2x sin 2 x lim 2 x →0 x lim
2
sin 2 x sin x 2 lim 2 = lim =1 =1 x →0 x → 0 x x sin 3 x lim x →0 4x sin 3 x sin 3 x 3 x 4x 3 3 lim = lim × × = 1× × 1 = x → 0 sin 4 x x →0 3x 4 x sin 4 x 4 4 1 − cos 2 x lim x →0 x2 1 − cos ( 2 x ) 1 − cos 2 x 1 lim = lim 4 × = 4× = 2 2 2 x →0 x → 0 x 2 ( 2x) lim x→
π
cos x π
2
−x
2
[email protected]
: 5:: ا )وال ا : , ا @)ت اE&9 اF* آ
lim x→
π cos 2 = lim t →0 t
cos x π
π
−x
2
= lim sin t = 1 t →0 t
−t
t=
:
π
−x
cd و.
x
"5b " ا55b7 م+
2
2
−5 lim + x 2 x →−∞ x −5 −5 lim + x 2 = lim + lim x 2 = 0 + ∞ = +∞ x →−∞ x →−∞ x x→−∞ x
lim x →1
x2 − 2 x x −1 −1 x2 − 2x 0 lim = x →1 x − 1 − 1 0
)د$ "5] O^_ lim+
x →1
lim−
x →1
x ( x − 2) x2 − 2 x x2 − 2 x x2 − 2 x = lim+ = lim+ = lim+ = lim+ x = 1 x →1 x →1 x − 1 − 1 x →1 x − 1 − 1 x →1 x − 2 x−2
x ( x − 2) x2 − 2 x x2 − 2 x x2 − 2x = lim− = lim+ = lim+ = lim+ − ( x − 2 ) = 1 x →1 x →1 x − 1 − 1 x →1 − x + 1 − 1 x →1 − x −x x2 − 2 x = 1 : إذن x −1 −1
lim x →1
x + 1 − x −1
lim
x →+∞
)د$ "5] O^_ lim
x →+∞
x + 1 − x − 1 = lim
(
x +1 − x −1
(
x →+∞
)(
lim
x →+∞
x +1 + x −1
x + 1 + x −1
)
x →1
x→1
)(
)
(
( x + 1) − ( x − 1) x +1 + x −1
)
= lim
x →+∞
(
2 x +1 + x −1
)
=
2 =0 +∞
3x + 1 − 2 x −1
)د$ "5] O^_
(
) = lim
x →+∞
lim
lim
x + 1 − x − 1 = +∞ − ∞
3x + 1 − 2 0 = x −1 0
lim x →1
3x+1−2 3x+1+2 3( x−1) 3x+1−2 3x+1−4 3x−3 3 3 =lim =lim =lim =lim =lim = x→1 x−1 x→1 ( x−1) 3x+1+2 ( x−1) 3x+1+2 x→1 ( x−1) 3x+1+2 x→1 ( x−1) 3x+1+2 x→1 3x+1+2 4
(
)
(
)
(
)
(
) (
)
π lim ( x − 1) tan x x →1 2
π
π
− t t −t 2 2 −2 π π π π 2 lim( x −1) tan x = limt tan ( t +1) = limt tan t + = lim = lim = lim− = x→1 t → 0 t → 0 t → 0 t → 0 t → 0 π π π π π π 2 2 2 2 tan t tan t tan t 2 2 2 2 lim x →0
lim+
x →0
( (
x2 + x x− x
) )
( (
) )
x x x +1 x x +1 x2 + x = lim = lim = −1 x →0 x − x x →0 x x − 1 x −1
: 5!eر إ : , *د ا @)ت ا1
[email protected]
x2 − 2 lim x →− 2 x + 2 x2 − 9 lim x →3 x − 3 2x −1 lim x →1+ − x + 1
lim 5 x 2 + 7 x − 2
sin 3 x x →0 7x sin 2 x lim x → 0 tan 2 x lim
x →3
3x 2 − 2 x + 1 x→2 x2 + 1 x2 + x − 2 lim x →1 x −1 lim
x +1 −1 x
lim x →0
5x − 1 2 x →+∞ x − 4 x + 5
lim − x 2 + 5 x + 4
lim
x →−∞
x4 + x + 1 x →−∞ 1 − x 5
2x −1 x →+∞ x − x + 2 lim
lim
2
"5;" ر
1 + x −1 : O^# ا ("! اf " ا )ا9( x 1 . x ∈ ℝ∗ O^ f ( x ) ≤ x أن57 ــ1 2 ؟g ــ ذا2 : O$ ا ∗ : )* . x ∈ ℝ '; ــ1 2
. f ( x) =
1 + x2 − 1 = x
f ( x) =
.
(
)(
1 + x2 − 1 x
(
)=
1 + x2 + 1
)
1 + x2 + 1
1 1 + x2 + 1
≤
x
(
1 + x2 − 1
)
1 + x2 + 1
=
x
=
1 + x2 + 1
x 1 + x2 + 1
1 و1 + x 2 + 1 ≥ 2 إذن1 + x 2 ≥ 1 )* x ∈ ℝ∗ :; ان7 و 2
. x ∈ ℝ∗ :; f ( x ) ≤
1 x و 2
x 1+ x2 + 1
. lim f ( x ) = 0 نlim x = 0 وf ( x ) ≤ x →0
x →0
≤
1 x " و 2
1 x ـــ أن2 2 : 2 =>" ر
. f ( x ) = 1 + x 2 − x : O^# ا ("! اf " ا )ا9( 1 أن57 ــ1 2x . lim f ( x ) g ــ &= ا2
. x ∈ ℝ ∗+ O^ f ( x ) ≤ x →+∞
f ( x ) = 1 + x2 − x =
.
(
1 1+ x + x 2
1+ x − x 2
)(
1+ x + x
1+ x + x 2
<
2
) = 1+ x − x 2
: O$ ا 2
1+ x + x 2
=
1 1+ x + x 2
: )* . x ∈ ℝ ∗+ ^5 ــ1
1 " و1 + x 2 + x > 2 x إذن1 + x 2 > x و1 + x 2 > x 2 نx ∈ ℝ ∗+ و أن 2x 1 . f ( x) < إذن 2x
[email protected]. 1 1 ــ ان2 . lim f ( x ) = 0 نlim = 0 وx ∈ ℝ ∗+ :; 0 < f ( x ) < x →+∞ x →+∞ 2 x 2x
: +"8 اhG 7 M.أ
: 3 =>" ر . f ( x) =
−2 x + x − 6 : O^# ا ("! اf " ا )ا9( x2 + 3 1 . x ∈ ℝ∗ O^ f ( x ) + 2 ≤ أن57 ــ1 x lim f ( x ) وlim f ( x ) g ــ &= ا2 2
x →−∞
x →+∞
: 4 =>" ر . f ( x ) = x sin x + 2 x : O^# ا ("! اf " ا )ا9( x ∈ ℝ ∗+ O^ f ( x ) > x أن57 ــ1
. lim f ( x ) g ــ &= ا2 x →+∞
[email protected]
