Ex Limite 1bacsc

'‫ا(ز‬

 ‫ر ا‬%‫& ا‬ !"#‫  ا‬$ : ‫إز اذ‬ ‫ 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"ال 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]