Limites trigonometricos1

Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos

1

Usar o limite fundamental e alguns artifícios : 1lim0

=→ x

senxx

1. x

xx senlim

0→= ? à

xx

x senlim

0→=

00 , é uma indeterminação.

xx

x senlim

0→=

xxx sen

1lim0→

=

xx

x

senlim

1

0→

= 1 logo x

xx senlim

0→= 1

2. x

xx

4senlim0→

= ? à x

xx

4senlim0→

=00 à

xx

x 44sen.4lim

0→= 4.

yy

y

senlim0→

=4.1= 4 logo

xx

x

4senlim0→

=4

3. x

xx 2

5senlim0→

= ? à =→ x

xx 5

5sen.25lim

0=

→ yy

y

sen.

25lim

0 25 logo

xx

x 25senlim

0→=

25

4. nx

mxx

senlim0→

= ? à nx

mxx

senlim0→

=mx

mxnm

x

sen.lim0→

=nm .

yy

y

senlim0→

=nm .1=

nm logo

nxmx

x

senlim0→

=nm

5. xx

x 2sen3senlim

0→= ? à

xx

x 2sen3senlim

0→= =

→

xx

xx

x 2sen

3sen

lim0

=→

xx

xx

x

22sen.2

33sen.3

lim0

.23

22senlim

33senlim

0

0 =

→

→

xx

xx

x

x . 1.23

senlim

senlim

0

0=

→

→

tt

yy

t

y =

23 logo

xx

x 2sen3senlim

0→=

23

6. sennxsenmx

x 0lim→

= ? à nxmx

x sensenlim

0→=

xnx

xmx

x sen

sen

lim0→

=

nxnxn

mxmxm

x sen.

sen.lim

0→=

nxnx

mxmx

nm

x sen

sen

.lim0→

=nm Logo

sennxsenmx

x 0lim→

=nm

7. =→ x

tgxx 0lim ? à =

→ xtgx

x 0lim

00 à =

→ xtgx

x 0lim =

→ xxx

xcossen

lim0

=→ xx

xx

1.cossenlim

0

xxx

x cos1.senlim

0→ =

xxx

xx cos1lim.senlim

00 →→ = 1 Logo =

→ xtgx

x 0lim 1

8. ( )11lim 2

2

1 −−

→ aatg

a= ? à ( )

11lim 2

2

1 −−

→ aatg

a=

00 à Fazendo

→→

−=01

,12

tx

at à ( )tttg

t 0lim→

=1

logo ( )11lim 2

2

1 −−

→ aatg

a=1


2

9. xxxx

x 2sen3senlim

0 +−

→= ? à

xxxx

x 2sen3senlim

0 +−

→=

00 à ( )

xxxxxf

2sen3sen

+−

= =

+

−

xxx

xxx

5sen1.

3sen1.=

+

−

xxx

xxx

.55sen.51.

.33sen.31.

=

xx

xx

.55sen.51

.33sen.31

+

− à

0lim→x

xx

xx

.55sen.51

.33sen.31

+

−=

5131

+− =

62− =

31

− logo

xxxx

x 2sen3senlim

0 +−

→=

31

−

10. 30

senlimx

xtgxx

−→

= ? à 30

senlimx

xtgxx

−→

= xx

xxx

xx cos1

1.sen.cos

1.senlim2

2

0 +→=

21

( )3

senx

xtgxxf

−= = 3

sencossen

x

xxx

−=

3cos

cos.sensen

xx

xxx −

= ( )xx

xxcos.

cos1.sen3

− =x

xxx

xcos

cos1.1.sen2

− =

xx

xx

xxx

cos1cos1.

coscos1.1.sen

2 ++− =

xxx

xxx

cos11.cos1.

cos1.sen

2

2

+− =

xxx

xxx

cos11.sen.

cos1.sen

2

2

+

Logo 30

senlimx

xtgxx

−→

=21

11. 30

sen11lim

xxtgx

x

+−+→

=? à xtgxx

xtgxx sen11

1.senlim 30 +++

−→

=

xtgxxxx

xxx

x sen111.

cos11.sen.

cos1.senlim 2

2

0 ++++→=

21.

21.

11.

11.1 =

41

( )3

11

x

senxtgxxf

+−+= =

xtgxxxtgx

sen111.sen11

3 +++

−−+ =xtgxx

xtgxsen11

1.sen3 +++

−

30

sen11lim

xxtgx

x

+−+→

=41

12. ax

axax −

−→

sensenlim = ? à ax

axax −

−→

sensenlim =

−

+

−

→

2.2

2cos.

2sen2

limax

axax

ax=

12

cos..

2.2

)2

sen(2lim

+

−

−

→

ax

ax

ax

ax= acos Logo

axax

ax −−

→

sensenlim = cosa


3

13. ( )a

xaxa

sensenlim0

−+→

= ? à ( )a

xaxa

sensenlim0

−+→

= 1

2cos.

.

2.2

2sen2

lim

++

−

−+

→

xax

ax

xax

aa=

12

2cos..

2.2

2sen2

lim

+

→

ax

a

a

aa= xcos Logo ( )

axax

a

sensenlim0

−+→

=cosx

14. ( )a

xaxa

coscoslim0

−+→

= ? à ( )a

xaxa

coscoslim0

−+→

= a

xaxxax

a

−−

++

−

→

2sen.

2sen2

lim0

=

−

−

+

−

→

2.2

2sen.

22sen.2

lim0 a

aax

a=

−

−

+

−→

2

2sen

.2

2senlim0 a

aax

a= xsen− Logo

( )a

xaxa

coscoslim0

−+→

=-senx

15. ax

axax −

−→

secseclim = ? à ax

axax −

−→

secseclim = ax

axax −

−

→

cos1

cos1

lim = ax

axxa

ax −

−

→

cos.coscoscos

lim =

( ) axaxxa

ax cos.cos.coscoslim

−−

→= ( ) axax

xaxa

ax cos.cos.2

sen.2

sen.2lim

−

−

+

−

→=

axxa

xaxa

ax cos.cos1.

2.2

2sen

.1

2sen.2

lim

−

−

−

+

−

→=

axxa

xaxa

ax cos.cos1.

2

2sen

.1

2sen

lim

−

−

+

→=

aaa

cos.cos1.1.

1sen =

aaa

cos1.

cossen = atga sec. Logo

axax

ax −−

→

secseclim = atga sec.

16. x

xx sec1lim

2

0 −→= ? à

xx

x sec1lim

2

0 −→=

( )xxxxx

cos11.

cos1.sen1lim

2

20

+−

→= 2−

( )

x

xxf

cos11

2

−= =

xx

x

cos1cos

2

−= ( )x

xxcos1.1

cos.2

−−= ( ) ( )

( )xx

xxx

cos1cos1.

cos1.cos1

1

2 ++−

− =

( )xxxx

cos11.

cos1.cos1

1

2

2

+−

−=

( )xxxx

cos11.

cos1.sen1

2

2

+−


4

17. tgx

gx

x −−

→ 1cot1

lim4π

= ? à tgx

gx

x −−

→ 1cot1lim

4π

= tgxtgx

x −

−

→ 1

11lim

4π

=tgx

tgxtgx

x −

−

→ 1

1

lim4π

=

tgxtgx

tgx

x −

−−

→ 1

)1.(1

lim4π

=tgxx

1lim4

−→

π= 1− Logo

tgxgx

x −−

→ 1cot1

lim4π

= -1

18. x

xx 2

3

0 sencos1lim −

→= ? à

xx

x 2

3

0 sencos1lim −

→= ( )( )

xxxx

x 2

2

0 cos1coscos1.cos1lim

−++−

→=

( )( )( )( )xx

xxxx cos1.cos1

coscos1.cos1lim2

0 +−++−

→=

xxx

x cos1coscos1lim

2

0 +++

→=

23 Logo

xx

x 2

3

0 sencos1lim −

→=

23

19. x

x

x cos.213senlim

3−→

π= ? à

xx

x cos.213senlim

3−→

π= ( )

1cos.21.senlim

3

xx

x

+−

→π

= 3−

( )x

xxfcos.213sen

−= = ( )

xxx

cos.212sen

−+ =

xxxxx

cos.21cos.2sen2cos.sen

−+ = ( )

xxxxxx

cos.21cos.cos.sen.21cos2.sen 2

−+− =

( )[ ]x

xxxcos.21

cos21cos2.sen 22

−+− = [ ]

xxx

cos.211cos4.sen 2

−− = ( )( )

xcoxcoxx

cos.21.21..21.sen

−+−

− = ( )1

cos.21.sen xx +−

20. tgx

xxx −

−→ 1

cossenlim4

π= ? à

tgxxx

x −−

→ 1cossenlim

4π

= ( )xx

coslim4

−→π

=22

−

( )tgx

xxxf−−

=1

cossen =

xx

xx

cossen1

cossen

−

− =

xx

xx

cossen1

cossen

−

− =

xxxxx

cossencoscossen

−− = ( )

xxx

xx

coscossen.1

cossen−−

− =

xxxxxsencos

cos.1

cossen−

−− = xcos−

21. ( ) )sec(cos.3lim3

xxx

π−→

= ? à ( ) )sec(cos.3lim3

xxx

π−→

= ∞.0

( ) ( ) )sec(cos.3 xxxf π−= = ( ) ( )xx

πsen1.3 − = ( )x

xππ −

−sen

3 = ( )xx

ππ −−

3sen3 = ( )

( )xx

−−

3.3sen.1

ππππ

=

( )( )x

xππ

πππ−

−3

3sen.1 à ( ) )sec(cos.3lim

3xx

xπ−

→= ( )

( )xxx

πππππ

−−→

33sen.1lim

3=

π1

22. )1sen(.limx

xx→∝

= ? à )1sen(.limx

xx→∝

= 0.∞

x

xx 1

1senlim

→∝= 1senlim

0=

→ tt

t à Fazendo

→+∞→

=0

1tx

xt


5

23. 1sen.3sen.2

1sensen.2lim 2

2

6 +−−+

→ xxxx

x π= ? à

1sen.3sen.21sensen.2lim 2

2

6 +−

−+→ xx

xxx π

=x

xx sen1

sen1lim6 +−

+→π

=

6sen1

6sen1

π

π

+−

+=

211

211

+−

+= 3− à ( )

1sen.3sen.21sensen.2

2

2

+−−+

=xx

xxxf =( )

( )1sen.21sen

1sen.21sen

−

−

+

−

xx

xx= ( )

( )1sen1sen

−+

xx =

xx

sen1sen1

+−+

24. ( )

−

→ 2.1lim

1

xtgxx

π = ? à ( )

−

→ 2.1lim

1

xtgxx

π = ∞.0 à ( ) ( )

−=

2.1 xtgxxf π =

( )

−−

22cot.1 xgx ππ = ( )

−

−

22

1xtg

xππ

=( )

−

−

22

2.1.2

xtg

x

πππ

π

=

( )x

xtg

−

−

1.2

22

2

π

πππ =

−

−

22

22

2

x

xtg

ππ

πππ à

( )

−

→ 2.1lim

1

xtgxx

π =

−

−

→

22

22

2

lim1

x

xtgx

ππ

πππ = ( )

tttg

t 0lim

2

→

π =π2 Fazendo uma mudança de variável,

temos :

→→

−=01

2 tx

xxt ππ

25. ( )xx

x πsen1lim

2

1

−→

= ? à ( )xx

x πsen1lim

2

1

−→

= ( )( )x

xx

x

πππππ

−−

+→ sen.

1lim1

=π2

( )x

xxfπsen

1 2−= = ( )( )

( )xxx

ππ −+−

sen1.1 = ( )

( )xx

x

−−

+

1sen

1ππ

= ( )( )x

xx

−−

+

1.sen.

1

ππππ

= ( )( )x

xx

πππππ

−−

+sen.

1

26.

−

→xgxg

x 2cot.2cotlim

0

π = ? à

−

→xgxg

x 2cot.2cotlim

0

π = 0.∞

( )

−= xgxgxf

2cot.2cot π = tgxxg .2cot =

xtgtgx

2=

xtgtgx

tgx

212−

= tgx

xtgtgx.2

1.2− =

21 2 xtg−

−

→xgxg

x 2cot.2cotlim

0

π =2

1lim2

0

xtgx

−→

=21

27. x

xxx 2

3

0 sencoscoslim −

→= 11102

2

1 ...1lim

ttttt

t +++++−

→=

121

−

( )x

xxxf 2

3

sencoscos −

= = 12

23

1 ttt

−

− = ( )( )( )11102

2

...1.11.

ttttttt

+++++−−− = 11102

2

...1 ttttt

+++++−

63.2 coscos xxt ==

→→

10

tx xt cos6 = , xt 212 cos= , 122 1sen tx −=


6

BriotxRuffini : 1 0 0 ... 0 -1 1 • 1 1 ... 1 1 1 1 1 ... 1 0

28. xxxx

x sencos12cos2senlim

4 −−−

→π= ? à

xxxx

x sencos12cos2senlim

4 −−−

→π= ( )x

xcos.2lim

4

−→π

= 4

cos.2 π− =

22.2− =

2−

( )xxxxxf

sencos12cos2sen

−−−

= = ( )xx

xxxsencos

11cos2cossen.2 2

−−−− =

xxxxx

sencos11cos2cos.sen.2 2

−−+− =

xxxxx

sencoscos2cos.sen.2 2

−− = ( )

xxxxx

sencossencos.cos.2

−−− = xcos.2−

29. ( )112

1senlim1 −−

−→ x

xx

= ? à ( )112

1senlim1 −−

−→ x

xx

= ( )( ) 1

112.1

1sen.21lim

1

+−−

−→

xx

xx

= 1

( ) ( )112

1sen−−

−=

xxxf = ( )

112112.

1121sen

+−

+−

−−

−

xx

xx = ( )

1112.

1121sen +−

−−− x

xx = ( )

( ) 1112.

1.21sen +−

−− x

xx =

( )( ) 1

112.1

1sen.21 +−

−− x

xx

30.

3

cos.21lim3

ππ−

−

→ x

x

x= ? à

3

cos.21lim3

ππ−

−

→ x

x

x=

−

−

+

→

23

23sen

.2

3sen.2lim3 x

x

x

x π

π

π

π=

.2

33sen.2

+ ππ= .

23

2sen.2

π= .

3sen.2

π = 3

23.2 =

( )

3

cos.21π

−

−=

x

xxf =

3

cos21.2

π−

−

x

x=

3

cos3

cos.2

π

π

−

−

x

x =

( )

−−

−

+−

23.2.1

23sen.

23sen2.2

x

xx

π

ππ

=

−

−

+

23

23sen.

23sen.2

x

xx

π

ππ

=

−

−

+

23

23sen

.2

3sen.2x

x

x

π

π

π

31. xxx

x sen.2cos1lim

0

−→

= ? à xxx

x sen.2cos1lim

0

−→

=x

x

x

sen.2lim3π

→= 2


7

( )xxxxf

sen.2cos1 −

= = ( )xx

xsen.

sen211 2−− =xx

xsen.

sen211 2+− =xxx

sen.sen.2 2

=x

xsen.2

32. xx

xx sen1sen1lim

0 −−+→= ? à

xxx

x sen1sen1lim

0 −−+→ =

xx

xxx sen.2

sen1sen1lim0

−++→

=1.211+

=1

( )xx

xxfsen1sen1 −−+

= = ( )( )xx

xxxsen1sen1

sen1sen1.−−+

−++ = ( )xx

xxxsen1sen1

sen1sen1.+−+

−++ =

( )x

xxxsen.2

sen1sen1. −++ =

xx

xxsen.2

sen1sen1 −++ = 1.211+ = 1

33. xx

xx sencos

2coslim0 −→

= 1

sencoslim0

xxx

+→

= 22

22

+ = 2

( )xx

xxfsencos2cos

−= = ( )

( )( )xxxxxxxsencos.sencos

sencos.2cos+−

+ = ( )xx

xxx22 sencossencos.2cos

−

+ = ( )x

xxx2cos

sencos.2cos + =

( )x

xxx2cos

sencos.2cos + = 1

sencos xx + = 22

22

+ = 2

34.

3

sen.23lim3

ππ−

−

→ x

x

x= ? à

3

sen.23lim3

ππ−

−

→ x

x

x=

3

sen23.2

lim3

ππ−

−

→ x

x

x=

3

sen3

sen.2lim

3π

π

π−

−

→ x

x

x=

3

23cos.

23sen.2

lim3

π

ππ

π−

+

−

→ x

xx

x=

33

23

3

cos.23

3

sen.2

lim3

π

ππ

π −

+

−

→ x

xx

x=

( )3

3.16

3cos.63sen.2

lim3

x

xx

x −−

+

−

→ π

ππ

π

35. ?

Limites trigonometricos1

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Transcript of Limites trigonometricos1