Limites trigonométricos

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Limites trigonométricos

  1. 1. Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos 1 Usar o limite fundamental e alguns artifícios : 1lim 0 = → x senx x 1. x x x sen lim 0→ = ? à x x x sen lim 0→ = 0 0 , é uma indeterminação. x x x sen lim 0→ = x xx sen 1 lim 0→ = x x x sen lim 1 0→ = 1 logo x x x sen lim 0→ = 1 2. x x x 4sen lim 0→ = ? à x x x 4sen lim 0→ = 0 0 à x x x 4 4sen .4lim 0→ = 4. y y y sen lim 0→ =4.1= 4 logo x x x 4sen lim 0→ =4 3. x x x 2 5sen lim 0→ = ? à = → x x x 5 5sen . 2 5 lim 0 = → y y y sen . 2 5 lim 0 2 5 logo x x x 2 5sen lim 0→ = 2 5 4. nx mx x sen lim 0→ = ? à nx mx x sen lim 0→ = mx mx n m x sen .lim 0→ = n m . y y y sen lim 0→ = n m .1= n m logo nx mx x sen lim 0→ = n m 5. x x x 2sen 3sen lim 0→ = ? à x x x 2sen 3sen lim 0→ = = → x x x x x 2sen 3sen lim 0 = → x x x x x 2 2sen .2 3 3sen .3 lim 0 . 2 3 2 2sen lim 3 3sen lim 0 0 = → → x x x x x x . 1. 2 3 sen lim sen lim 0 0 = → → t t y y t y = 2 3 logo x x x 2sen 3sen lim 0→ = 2 3 6. sennx senmx x 0 lim → = ? à nx mx x sen sen lim 0→ = x nx x mx x sen sen lim 0→ = nx nx n mx mx m x sen . sen . lim 0→ = nx nx mx mx n m x sen sen .lim 0→ = n m Logo sennx senmx x 0 lim → = n m 7. = → x tgx x 0 lim ? à = → x tgx x 0 lim 0 0 à = → x tgx x 0 lim = → x x x x cos sen lim 0 = → xx x x 1 . cos sen lim 0 xx x x cos 1 . sen lim 0→ = xx x xx cos 1 lim. sen lim 00 →→ = 1 Logo = → x tgx x 0 lim 1 8. ( ) 1 1 lim 2 2 1 − − → a atg a = ? à ( ) 1 1 lim 2 2 1 − − → a atg a = 0 0 àFazendo    → → −= 0 1 ,12 t x at à ( ) t ttg t 0 lim → =1 logo ( ) 1 1 lim 2 2 1 − − → a atg a =1
  2. 2. Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos 2 9. xx xx x 2sen 3sen lim 0 + − → = ? à xx xx x 2sen 3sen lim 0 + − → = 0 0 à ( ) xx xx xf 2sen 3sen + − = =       +       − x x x x x x 5sen 1. 3sen 1. =       +       − x x x x x x .5 5sen .51. .3 3sen .31. = x x x x .5 5sen .51 .3 3sen .31 + − à 0 lim →x x x x x .5 5sen .51 .3 3sen .31 + − = 51 31 + − = 6 2− = 3 1 − logo xx xx x 2sen 3sen lim 0 + − → = 3 1 − 10. 30 sen lim x xtgx x − → = ? à 30 sen lim x xtgx x − → = xx x xx x x cos1 1 . sen . cos 1 . sen lim 2 2 0 +→ = 2 1 ( ) 3 sen x xtgx xf − = = 3 sen cos sen x x x x − = 3 cos cos.sensen x x xxx − = ( ) xx xx cos. cos1.sen 3 − = x x xx x cos cos1 . 1 . sen 2 − = x x x x xx x cos1 cos1 . cos cos1 . 1 . sen 2 + +− = xx x xx x cos1 1 . cos1 . cos 1 . sen 2 2 + − = xx x xx x cos1 1 . sen . cos 1 . sen 2 2 + Logo 30 sen lim x xtgx x − → = 2 1 11. 30 sen11 lim x xtgx x +−+ → =? à xtgxx xtgx x sen11 1 . sen lim 30 +++ − → = xtgxxx x xx x x sen11 1 . cos1 1 . sen . cos 1 . sen lim 2 2 0 ++++→ = 2 1 . 2 1 . 1 1 . 1 1 .1 = 4 1 ( ) 3 11 x senxtgx xf +−+ = = xtgxx xtgx sen11 1 . sen11 3 +++ −−+ = xtgxx xtgx sen11 1 . sen 3 +++ − 30 sen11 lim x xtgx x +−+ → = 4 1 12. ax ax ax − − → sensen lim = ? à ax ax ax − − → sensen lim =       −       +       − → 2 .2 2 cos. 2 sen2 lim ax axax ax = 1 2 cos. . 2 .2 ) 2 sen(2 lim       +       − − → ax ax ax ax = acos Logo ax ax ax − − → sensen lim = cosa
  3. 3. Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos 3 13. ( ) a xax a sensen lim 0 −+ → = ? à ( ) a xax a sensen lim 0 −+ → = 1 2 cos. . 2 .2 2 sen2 lim       ++       −       −+ → xax ax xax aa = 1 2 2 cos. . 2 .2 2 sen2 lim       +             → ax a a aa = xcos Logo ( ) a xax a sensen lim 0 −+ → =cosx 14. ( ) a xax a coscos lim 0 −+ → = ? à ( ) a xax a coscos lim 0 −+ → = a xaxxax a       −−       ++ − → 2 sen. 2 sen2 lim 0 =       −       −       + − → 2 .2 2 sen. 2 2 sen.2 lim 0 a aax a =       −       −       + − → 2 2 sen . 2 2 senlim 0 a a ax a = xsen− Logo ( ) a xax a coscos lim 0 −+ → =-senx 15. ax ax ax − − → secsec lim = ? à ax ax ax − − → secsec lim = ax ax ax − − → cos 1 cos 1 lim = ax ax xa ax − − → cos.cos coscos lim = ( ) axax xa ax cos.cos. coscos lim − − → = ( ) axax xaxa ax cos.cos. 2 sen. 2 sen.2 lim −       −       + − → = axxa xaxa ax cos.cos 1 . 2 .2 2 sen . 1 2 sen.2 lim       − −       −       + − → = axxa xaxa ax cos.cos 1 . 2 2 sen . 1 2 sen lim       −       −       + → = aa a cos.cos 1 .1. 1 sen = aa a cos 1 . cos sen = atga sec. Logo ax ax ax − − → secsec lim = atga sec. 16. x x x sec1 lim 2 0 −→ = ? à x x x sec1 lim 2 0 −→ = ( )xxx xx cos1 1 . cos 1 . sen 1 lim 2 20 + − → = 2− ( ) x x xf cos 1 1 2 − = = x x x cos 1cos 2 − = ( )x xx cos1.1 cos.2 −− = ( ) ( ) ( )x x xx x cos1 cos1 . cos 1 . cos1 1 2 + +− − = ( )xxx x cos1 1 . cos 1 . cos1 1 2 2 + − − = ( )xxx x cos1 1 . cos 1 . sen 1 2 2 + −
  4. 4. Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos 4 17. tgx gx x − − → 1 cot1 lim 4 π = ? à tgx gx x − − → 1 cot1 lim 4 π = tgx tgx x − − → 1 1 1 lim 4 π = tgx tgx tgx x − − → 1 1 lim 4 π = tgx tgx tgx x − −− → 1 )1.(1 lim 4 π = tgxx 1 lim 4 − → π = 1− Logo tgx gx x − − → 1 cot1 lim 4 π = -1 18. x x x 2 3 0 sen cos1 lim − → = ? à x x x 2 3 0 sen cos1 lim − → = ( )( ) x xxx x 2 2 0 cos1 coscos1.cos1 lim − ++− → = ( )( ) ( )( )xx xxx x cos1.cos1 coscos1.cos1 lim 2 0 +− ++− → = x xx x cos1 coscos1 lim 2 0 + ++ → = 2 3 Logo x x x 2 3 0 sen cos1 lim − → = 2 3 19. x x x cos.21 3sen lim 3 −→ π = ? à x x x cos.21 3sen lim 3 −→ π = ( ) 1 cos.21.sen lim 3 xx x + − → π = 3− ( ) x x xf cos.21 3sen − = = ( ) x xx cos.21 2sen − + = x xxxx cos.21 cos.2sen2cos.sen − + = ( ) x xxxxx cos.21 cos.cos.sen.21cos2.sen 2 − +− = ( )[ ] x xxx cos.21 cos21cos2.sen 22 − +− = [ ] x xx cos.21 1cos4.sen 2 − − = ( )( ) x coxcoxx cos.21 .21..21.sen − +− − = ( ) 1 cos.21.sen xx + − 20. tgx xx x − − → 1 cossen lim 4 π = ? à tgx xx x − − → 1 cossen lim 4 π = ( )x x coslim 4 − →π = 2 2 − ( ) tgx xx xf − − = 1 cossen = x x xx cos sen 1 cossen − − = x x xx cos sen 1 cossen − − = x xx xx cos sencos cossen − − = ( ) x xx xx cos cossen.1 cossen −− − = xx xxx sencos cos . 1 cossen − − − = xcos− 21. ( ) )sec(cos.3lim 3 xx x π− → = ? à ( ) )sec(cos.3lim 3 xx x π− → = ∞.0 ( ) ( ) )sec(cos.3 xxxf π−= =( ) ( )x x πsen 1 .3 − = ( )x x ππ − − sen 3 = ( )x x ππ − − 3sen 3 = ( ) ( )x x − − 3. 3sen. 1 π πππ = ( ) ( )x x ππ πππ − − 3 3sen. 1 à ( ) )sec(cos.3lim 3 xx x π− → = ( ) ( )x xx ππ πππ − −→ 3 3sen. 1 lim 3 = π 1 22. ) 1 sen(.lim x x x→∝ = ? à ) 1 sen(.lim x x x→∝ = 0.∞ x x x 1 1 sen lim       →∝ = 1 sen lim 0 = → t t t à Fazendo    → +∞→ = 0 1 t x x t
  5. 5. Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos 5 23. 1sen.3sen.2 1sensen.2 lim 2 2 6 +− −+ → xx xx x π = ? à 1sen.3sen.2 1sensen.2 lim 2 2 6 +− −+ → xx xx x π = x x x sen1 sen1 lim 6 +− + →π = 6 sen1 6 sen1 π π +− + = 2 1 1 2 1 1 +− + = 3− à ( ) 1sen.3sen.2 1sensen.2 2 2 +− −+ = xx xx xf = ( ) ( )1sen. 2 1 sen 1sen. 2 1 sen −      − +      − xx xx = ( ) ( )1sen 1sen − + x x = x x sen1 sen1 +− + 24. ( )       − → 2 .1lim 1 x tgx x π = ? à ( )       − → 2 .1lim 1 x tgx x π = ∞.0 à ( ) ( )       −= 2 .1 x tgxxf π = ( )       −− 22 cot.1 x gx ππ = ( )       − − 22 1 x tg x ππ = ( )       − − 22 2 .1. 2 x tg x ππ π π = ( )x x tg −       − 1. 2 22 2 π ππ π =       −       − 22 22 2 x x tg ππ ππ π à ( )       − → 2 .1lim 1 x tgx x π =       −       − → 22 22 2 lim 1 x x tg x ππ ππ π = ( ) t ttg t 0 lim 2 → π = π 2 Fazendo uma mudança de variável, temos :    → → −= 0 1 2 t x x x t ππ 25. ( )x x x πsen 1 lim 2 1 − → = ? à ( )x x x πsen 1 lim 2 1 − → = ( ) ( )x x x x ππ πππ − − + → sen. 1 lim 1 = π 2 ( ) x x xf πsen 1 2 − = = ( )( ) ( )x xx ππ − +− sen 1.1 = ( ) ( )x x x − − + 1 sen 1 ππ = ( ) ( )x x x − − + 1. sen. 1 π πππ = ( ) ( )x x x ππ πππ − − + sen. 1 26.       − → xgxg x 2 cot.2cotlim 0 π = ? à       − → xgxg x 2 cot.2cotlim 0 π = 0.∞ ( )       −= xgxgxf 2 cot.2cot π = tgxxg .2cot = xtg tgx 2 = xtg tgx tgx 2 1 2 − = tgx xtg tgx .2 1 . 2 − = 2 1 2 xtg−       − → xgxg x 2 cot.2cotlim 0 π = 2 1 lim 2 0 xtg x − → = 2 1 27. x xx x 2 3 0 sen coscos lim − → = 11102 2 1 ...1 lim tttt t t +++++ − → = 12 1 − ( ) x xx xf 2 3 sen coscos − = = 12 23 1 t tt − − = ( ) ( )( )11102 2 ...1.1 1. ttttt tt +++++− −− = 11102 2 ...1 tttt t +++++ − 63.2 coscos xxt ==    → → 1 0 t x xt cos6 = , xt 212 cos= , 122 1sen tx −=
  6. 6. Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos 6 BriotxRuffini : 1 0 0 ... 0 -1 1 • 1 1 ... 1 1 1 1 1 ... 1 0 28. xx xx x sencos 12cos2sen lim 4 − −− →π = ? à xx xx x sencos 12cos2sen lim 4 − −− →π = ( )x x cos.2lim 4 − →π = 4 cos.2 π − = 2 2 .2− = 2− ( ) xx xx xf sencos 12cos2sen − −− = = ( ) xx xxx sencos 11cos2cossen.2 2 − −−− = xx xxx sencos 11cos2cos.sen.2 2 − −+− = xx xxx sencos cos2cos.sen.2 2 − − = ( ) xx xxx sencos sencos.cos.2 − −− = xcos.2− 29. ( ) 112 1sen lim 1 −− − → x x x = ? à ( ) 112 1sen lim 1 −− − → x x x = ( ) ( ) 1 112 . 1 1sen . 2 1 lim 1 +− − − → x x x x = 1 ( ) ( ) 112 1sen −− − = x x xf = ( ) 112 112 . 112 1sen +− +− −− − x x x x = ( ) 1 112 . 112 1sen +− −− − x x x = ( ) ( ) 1 112 . 1.2 1sen +− − − x x x = ( ) ( ) 1 112 . 1 1sen . 2 1 +− − − x x x 30. 3 cos.21 lim 3 ππ − − → x x x = ? à 3 cos.21 lim 3 ππ − − → x x x =         −         −         + → 2 3 2 3sen . 2 3sen.2lim 3 x x x x π π π π = . 2 33sen.2         +ππ = . 2 3 2 sen.2         π = . 3 sen.2      π = 3 2 3 .2 = ( ) 3 cos.21 π − − = x x xf = 3 cos 2 1 .2 π −       − x x = 3 cos 3 cos.2 π π −       − x x = ( )         − −         −         + − 2 3.2.1 2 3sen. 2 3sen2.2 x xx π ππ =         −         −         + 2 3 2 3sen. 2 3sen.2 x xx π ππ =         −         −         + 2 3 2 3sen . 2 3sen.2 x x x π π π 31. xx x x sen. 2cos1 lim 0 − → = ? à xx x x sen. 2cos1 lim 0 − → = x x x sen.2 lim 3 π → = 2
  7. 7. Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos 7 ( ) xx x xf sen. 2cos1− = = ( ) xx x sen. sen211 2 −− = xx x sen. sen211 2 +− = xx x sen. sen.2 2 = x xsen.2 32. xx x x sen1sen1 lim 0 −−+→ = ? à xx x x sen1sen1 lim 0 −−+→ = x x xx x sen.2 sen1sen1 lim 0 −++ → = 1.2 11+ =1 ( ) xx x xf sen1sen1 −−+ = = ( ) ( )xx xxx sen1sen1 sen1sen1. −−+ −++ = ( ) xx xxx sen1sen1 sen1sen1. +−+ −++ = ( ) x xxx sen.2 sen1sen1. −++ = x x xx sen .2 sen1sen1 −++ = 1.2 11+ = 1 33. xx x x sencos 2cos lim 0 −→ = 1 sencos lim 0 xx x + → = 2 2 2 2 + = 2 ( ) xx x xf sencos 2cos − = = ( ) ( )( )xxxx xxx sencos.sencos sencos.2cos +− + = ( ) xx xxx 22 sencos sencos.2cos − + = ( ) x xxx 2cos sencos.2cos + = ( ) x xxx 2cos sencos.2cos + = 1 sencos xx + = 2 2 2 2 + = 2 34. 3 sen.23 lim 3 ππ − − → x x x = ? à 3 sen.23 lim 3 ππ − − → x x x = 3 sen 2 3 .2 lim 3 ππ −         − → x x x = 3 sen 3 sen.2 lim 3 π π π −       − → x x x = 3 2 3cos. 2 3sen.2 lim 3 π ππ π −                         +             − → x xx x = 3 3 2 3 3 cos. 2 3 3 sen.2 lim 3 π ππ π −                         +             − → x xx x = ( ) 3 3.1 6 3 cos. 6 3 sen.2 lim 3 x xx x −−             +       − → π ππ π 35. ?

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