limitele_functiilor_elementare.docx

8/19/2019 limitele_functiilor_elementare.docx

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Limitele func iilor elementareț Aplica iiț

1) Func ia constantăț

,: R R f → c x f =)(

, c∈

R.

=→

)(lim x f a x

c,

Ra∈∀

==→→12lim)(lim

22 x x x f

==−∞→−∞→12lim)(lim

x x x f

==∞→∞→12lim)(lim

x x x f

2) Func ia polinomialăț

});2,1{,0( ∈≠ nan ,: R R f →

01

1

1 ...)( a xa xa xa x f n

n

n

n ++++= −−

,

ni Rai ,0, =∈,

;),()(lim Raa f x f a x

∈=→

∞−

−>−<∞==

−∞→−∞→ .,

);,0(),0(,lim)(lim

înrest

par na sauimpar nadacăa x x f

nn

n

n

x x

<∞− >∞== ∞→∞→ .0,

;0,lim)(limn

nn

n

x x dacăadacăaa x x f

=+−→ )143(lim

2

2 x x x

=+−−∞→

)143(lim 2 x x x

=+−∞→

)143(lim 2 x x x

3) Func ia ra ionalăț ț

,}0)(|{\: R xQ x R f →=

01

1

1

01

1

1

...

...

)(

)()(

b xb xb xb

a xa xa xa

xQ

x P x f

m

m

m

m

n

n

n

n

++++++++

== −−

−−

;

;0)(,),()(lim ≠∈=→

aQ Raa f x f a x

dacă x = a i Q(a) = 0 se calculează limieleș

lae!ale;

>±∞

=

<

=

−

±∞→

.:,)(

;:,

;:,0

)(lim

mndacăb

a

mndacăb

a

mndacă

x f

mn

m

n

m

n

x

=++−

→ 4

13lim

2

2 x

x x

x

=+−+−

∞→ 32

13lim

3

2

x x

x x

x

=++−

−∞→ 2"

13lim

2

2

x

x x

x

=+−+−

−∞→ 4"

13lim

2

x

x x

x

4) Func ia radicalț

,),0#: R f →∞ x x f =)(

;

;0,)(lim ≥=→

aa x f a x

.lim ∞=∞→

x x

,: R R f → 3)( x x f =;

;lim 33 a xa x=

→−∞=

−∞→

3lim x x

;

.lim 3 ∞=∞→

x x

Teoremă $ie i!ul (xș

n

), cu

.,1, Rnn xn ∈≥= α α

=−→

12lim3

x x

=+∞→

13lim x x



%u&ci

>∞=<

=∞→

.0,

,0,1

,0.,0

lim

α

α

α

α

dacă

dacă

dacă

nn

=−+−−∞→

3 3 12lim x x x

5) Func ia exponen ialăț ț

),,0(: ∞→ R f .1,0,)( ≠>= bbb x f x

'acă 0 1:

;,lim Rabb a x

a x∈=

→;lim ∞=

−∞→

x

xb .0lim =

∞→

x

xb

'acă 1 :

;,lim Rabb a xa x ∈=→ ;0lim =−∞→

x x b .lim ∞=∞→

x x b

Teoremă $ie i!ul (aș

n

), cu aza a.

%u&ci

−≤>∞=−∈

=∞→

.1,

,1,

,1,1

),1,1(,0

lim

dacăanuexistă

dacăa

dacăa

dacăa

a n

n

=→

x

x"lim

3

;

→

x

x *

2lim

3

=−∞→

x

x 3lim ;

−∞→ x "

3lim

=++

∞→ x x

x x

x 43

32lim

=∞→ x x

x

2lim

2

6. Func ia logaritmicăț +: (0, ,) R→∞+(x) =

xbl-,

.1,0 ≠> bb

'acă 0 1:

;0,)(lim

00

=∞=>→

a x f

x x );,0(),()(lim ∞∈=

→aa f x f

a x

.)(lim −∞=∞→

x f x

'acă 1 :

;)(lim

00

−∞=

>

→

x f

x x );,0(),()(lim ∞∈=

→aa f x f

a x.)(lim ∞=

∞→ x f

x

=

>

→

x

x x

l-lim

00

;

>→

x x

00

l-lim

=∞→

x x

2

1l-lim

;

>

→

x x

00llim



. Limitele func iilor trigonometriceț

,1,1#: −→ R f x x f si&)( =;

a xa x

si&si&lim =→

;

,1,1#: −→ R f x x f cs)( =;

a xa x

cscslim =→

;

( ) ,}|2

12{\: R Z k k R f →∈+ π

tgx x f =)(

;

tgatgxa x=

→lim

,

( ) }2

12{\ π

+∈ k Ra

;

−∞=

>

→ tgx x x

2/2/lim

π

π

;

∞=<→ tgx

x x

2/2/lim

π

π

,}|{\: R Z k k R f →∈π ctgx x f =)(

;

ctgactgxa x=

→lim

,

}|{\ Rk k Ra ∈∈ π

;

∞=

>

→

ctgx

x x

00

lim

;

−∞=

<

→

ctgx

x x

00

lim

.

,2,21,1#:

−→−

π π

f x x f a!csi&)( =;

a xa x a!csi&a!csi&lim =→

;

[ ],,01,1#: π →− f x x f a!ccs)( =;

a xa x

a!ccsa!ccslim =→

;

,2

,2

:

−→

π π R f

arctgx x f =)(

;

arctgaarctgxa x=

→lim

,

Ra∈;

2lim

π =

∞→arctgx

x

;

2lim

π −=

−∞→arctgx

x

.

),,0(: π → R f arcctgx x f =)(

;

arcctgaarcctgxa x=

→lim

,

Ra∈;

π =−∞→arcctgx

xlim

;

0lim =∞→arcctgx

x

.

=→

x x

si&lim

2

π

;

→ x

slim

4

π

=→

x x

cslim

3

π

;

→ xclim

4

π

=→ tgx x3

limπ

;

→ tg x4

limπ

=→

ctgx x

3

limπ

;

→c

x4

limπ

=−→

x x

a!csi&lim1

;

→ x

lim

2

2

=→

x x

a!ccslim

2

1

;

→ x

alim

2

2

=→

arctgx x 3lim

;

→ar

x 1lim

=→

arcctgx x 3lim

;

→ar

x 1lim

L!"!T# $#"A$%A&!L#

( ) et x x

t t

x

x

x

x=+=

+=

+

→−∞→∞→

1

01lim11lim11lim

;

( )1

1l&lim

0=+

→ x

x

x

;

( )1,0,l&1

lim0

≠>=−

→aaa

x

a x

x

;

=

+∞→

x

x x2

11lim

( ) =+→

x x

x1

0si&1lim

( )=+

→ x

x

x

"1l&lim

0

=−

→ x

e x

x 3

1lim

0



( ).,

11lim

0 Rr r

x

x r

x∈=

−+→

1si&

lim0=

→ x

x

x

;

1lim0=

→ x

tgx

x

;

1a!csi&

lim0=

→ x

x

x

;

1lim0=

→ x

arctgx

x

.

=−+

→ x

x

x

1)41(lim

2

0

=→ x

x

x

4si&lim

0

=→ x

xtg

x 2

3lim

0

=→ x

x

x

"a!csi&

lim0 → x

xarctg

x 2

3

lim0

'pera ii cu limite de func iiț ț

Teoremă $ie R E g f →:, , E a∈ iș Rc∈ , ia!

1)(lim l x f a x

=→

,

2)(lim l x g a x

=→

. %u&ci:

1)

=+→

))((lim x g f a x

+→

)(lim x f a x

21)(lim l l x g a x+=

→

; az excea:∞−∞

;

2)

c xcf a x=

→)(lim =→

)(lim x f a x 1cl

;

3)

=⋅→

))((lim x g f a x

⋅→

)(lim x f a x

21)(lim l l x g a x⋅=

→

; az excea:

( )∞±⋅0

;

4)

=→ )(

)(lim

x g

x f

a x

=

→

→

a x

a x

x g

x f

)(lim

)(lim

2

1

l

l

; azu!i exceae:∞±∞±

;0

0

;

")

=→

|)(|lim x f a x

|

)(lim x f a x→

| = |1l

|;

) dacă0)( > x f

, E x∈∀ i dacăș

2

1

l

l a!e se&s, au&ci:

=→

)())((lim

x g

a x

x f 2

1

)(lim

)(lim l

x g

a x

l x f a x =→→

;

azu!i a!icula!e:

nna x

n

a xl x f x f 1)(lim)(lim ==

→→

;

2)(lim

)(lim l

x g x g

a xbbb a x == →

→

, 5 0;

azu!i exceae:

∞±∞ 1;;0 00

.



Aplica ii(ț

1)

)l&"3(lim1

x x x

x++

→

; 2) x x

x

x −+−

∞→ 2

2

"

2lim

; 3)2

4lim

2

2 −−

→ x

x

x

; 4)

x

x

x 41

3lim

+−∞→

;

")

))(cs(si&lim

3

x x x

π →

; )

x

x

x 3...331

2...221lim

2

2

++++++++

∞→

; *)

x x

x

x ++∞→ 1lim

2

; 6)21

lim+++∞→ x x

x

x

;

7)

12

3

1lim

+

∞→

+

x

x

x x

x

; 10)

1

2

2

4

1lim

+

∞→

++

x

x x x

x

.

limitele_functiilor_elementare.docx

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