Limites trigonometricos1

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Limites trigonometricos1

  1. 1. Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos senxUsar o limite fundamental e alguns artifícios : lim =1 x→0 x x x 01. lim =? à lim = , é uma indeterminação. x → 0 sen x x →0 sen x 0 x 1 1 xlim = lim = = 1 logo lim =1x →0 sen x x →0 sen x sen x x → 0 sen x lim x x→0 x sen 4 x sen 4 x 0 sen 4 x sen y2. lim = ? à lim = à lim 4. = 4. lim =4.1= 4 logo x→0 x x→0 x 0 x →0 4x y →0 y sen 4 x lim =4 x→0 x sen 5 x 5 sen 5 x 5 sen y 5 sen 5 x 53. lim = ? à lim . = lim . = logo lim = x →0 2 x x →0 2 5x y →0 2 y 2 x →0 2 x 2 sen mx sen mx m sen mx m sen y m m sen mx m4. lim = ? à lim = lim . = . lim = .1= logo lim = x →0 nx x →0 nx x→0 n mx n y →0 y n n x →0 nx n sen 3 x sen 3 x sen 3 x sen y 3. lim lim sen 3 x sen 3 x x →0 3 x 3 y →0 y 35. lim =? à lim = lim x = lim 3x = . = . = .1 = x →0 sen 2 x x →0 sen 2 x x → 0 sen 2 x x→0 sen 2 x sen 2 x 2 sen t 2 2. lim lim x 2x x→0 2 x t →0 t 3 sen 3 x 3 logo lim = 2 x →0 sen 2 x 2 sen mx sen mx sen mx m. senmx sen mx x mx = lim m . mx = m6. lim = ? à lim = lim = lim Logo x→0 sennx x →0 sen nx x → 0 sen nx x →0 sen nx x →0 n sen nx n n. x nx nx senmx m lim = x → 0 sennx n sen x tgx tgx 0 tgx sen x 17. lim = ? à lim = à lim = lim cos x = lim . = x→ 0 x x→ 0 x 0 x→ 0 x x→ 0 x x → 0 cos x x sen x 1 sen x 1 tgx lim . = lim . lim = 1 Logo lim =1 x→ 0 x cos x x→ 0 x x → 0 cos x x→ 0 x8. lim ( tg a 2 − 1 ) = ? à lim 2 tg a 2 − 1 (= 0 ) x → 1 à Fazendo t = a 2 − 1,  à lim tg (t ) =1 a →1 a − 1 2 a →1 a − 1 0 t →0 t →0 t logo lim ( ) =1 tg a 2 − 1 a →1 a2 −1 1
  2. 2. Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos  sen 3 x  x.1 −  x − sen 3 x x − sen 3 x x − sen 3 x  x  à f (x ) = 09. lim = ? à lim = = = x →0 x + sen 2 x x →0 x + sen 2 x 0 x + sen 2 x  sen 5 x  x.1 +   x   sen 3 x  sen 3 x sen 3 x x.1 − 3.  1 − 3. 1 − 3.  3. x  = 3. x à lim 3.x = 1 − 3 = −2 = − 1 logo  sen 5 x  sen 5 x x →0 sen 5 x 1+ 5 6 3 x.1 + 5.  1 + 5. 1 + 5.  5. x  5. x 5. x x − sen 3 x 1 lim =− x →0 x + sen 2 x 3 tgx − sen x tgx − sen x sen x 1 sen 2 x 1 110. lim = ? à lim = lim . . . = x →0 x 3 x →0 x 3 x→0 x cos x x 2 1 + cos x 2 sen x sen x − sen x. cos x − sen x tgx − sen x cos x sen x.(1 − cos x ) sen x 1 1 − cos x f (x ) = = = cos x = = . . = x 3 x3 x3 x 3 . cos x x x 2 cos xsen x 1 1 − cos x 1 + cos x sen x 1 1 − cos 2 x 1 sen x 1 sen 2 x 1 . . . = . . . = . . . x x 2 cos x 1 + cos x x cos x x 2 1 + cos x x cos x x 2 1 + cos x tgx − sen x 1Logo lim = x →0 x3 2 1 + tgx − 1 + sen x tgx − sen x 111. lim =? à lim . = x →0 x 3 x →0 x 3 1 + tgx + 1 + sen x sen x 1 sen 2 x 1 1 1 1 1 1 1lim . . . . = 1. . . . =x →0 x cos x x 2 1 + cos x 1 + tgx + 1 + sen x 1 1 2 2 4 1 + tgx − 1 + senx 1 + tgx − 1 − sen x tgx − sen x f (x ) = 1 1 = . = . x3 x3 1 + tgx + 1 + sen x x3 1 + tgx + 1 + sen x 1 + tgx − 1 + sen x 1lim 3 =x →0 x 4 x−a x+a 2 sen . cos  sen x − sen a sen x − sen a  2   2 =12. lim =? à lim = lim x→a x−a x→a x−a x→a x−a 2.   2  x − a . cos x + a    2 sen( ) 2 .  2  sen x − sen a lim = cos a Logo lim = cosa x→a x−a 1 x→a x−a 2.   2  2
  3. 3. Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos  x+a−x  x+a+ x 2 sen  . cos  sen ( x + a ) − sen x sen ( x + a ) − sen x  2   2 13. lim = ? à lim = lim . = a →0 a a →0 a a→a  x−a 1 2.   2  a  2x + a  2 sen  . cos   2  2  sen ( x + a ) − sen x lim . = cos x Logo lim =cosx a→a a 1 a →0 a 2.  2 x+a+ x x−a− x − 2 sen . sen  cos( x + a ) − cos x cos( x + a ) − cos x  2   2 14. lim = ? à lim = lim = a→0 a a→0 a a →0 a  2x + a  −a −a − 2. sen . sen  sen  2x + a  lim  2   2  = lim − sen  .  2  = − sen x Logo a→0 −a a→0  2  −a 2.     2   2  cos( x + a ) − cos x lim =-senx a→0 a 1 1 cos a − cos x − sec x − sec a sec x − sec a cos x cos a15. lim = ? à lim = lim = lim cos x. cos a = x→a x−a x→a x−a x→ a x−a x→a x−a a+ x a−x − 2. sen . sen   cos a − cos x  2   2 = lim = lim x → a ( x − a ). cos x. cos a x→a (x − a ). cos x. cos a a+ x a−x a+x a−x − 2. sen   sen  sen  sen   2 .  2 . 1  2   2  1 lim = lim . . = x→a 1  a − x  cos x. cos a x→ a 1  a − x  cos x. cos a − 2.     2   2  sen a 1 sen a 1 sec x − sec a .1 . = . = tga. sec a Logo lim = tga. sec a 1 cos a. cos a cos a cos a x→a x−a16. lim x2 x → 0 1 − sec x = ? à lim x2 x → 0 1 − sec x = lim x→0 2 1 = − 2 sen x 1 1 − . . x2 cos x (1 + cos x ) x2 x2 x 2 . cos xf (x ) = 1 = = = = 1− 1 cos x − 1 − 1.(1 − cos x ) − (1 − cos x ) . 1 . (1 + cos x ) cos x cos x x2 cos x (1 + cos x ) 1 1 = 1 − cos 2 x 1 1 sen 2 x 1 1− . . − . . x 2 cos x (1 + cos x ) x 2 cos x (1 + cos x ) 3
  4. 4. Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos 1 tgx − 1 1− 1 − cot gx 1 − cot gx tgx tgx17. lim =?à lim = lim = lim = π 1 − tgx π 1 − tgx x → 1 − tgx π π 1 − tgx x→ x→ x→ 4 4 4 4 −1.(1 − tgx ) tgx 1 1 − cot gx lim = lim − = −1 Logo lim = -1 x→ π 1 − tgx π x→ tgx π x→ 1 − tgx 4 4 418. lim 1 − cos x 3 = ? à lim 1 − cos x 3 = lim (1 − cos x ).(1 + cos x + cos 2 x ) = x→0 sen 2 x x→0 sen 2 x x →0 1 − cos 2 x lim (1 − cos x ).(1 + cos x + cos 2 x ) = lim 1 + cos x + cos 2 x = 3 Logo lim 1 − cos 3 x = 3 x →0 (1 − cos x )(1 + cos x ) . x →0 1 + cos x 2 x → 0 sen 2 x 2 sen 3 x sen 3 x sen x.(1 + 2. cos x )19. lim = ? à lim = lim − =− 3 π x→ 1 − 2. cos x π x→ 1 − 2. cos x π x→ 1 3 3 3 f (x ) = sen 3 x = sen ( x + 2 x ) = sen x. cos 2 x + sen 2 x. cos x = ( sen x. 2 cos 2 x − 1 + 2. sen x. cos x. cos x = ) 1 − 2. cos x 1 − 2. cos x 1 − 2. cos x 1 − 2. cos x [( )sen x. 2 cos 2 x − 1 + 2 cos 2 x = ] sen x. 4 cos 2 x − 1 =− [ ] sen x.(1 − 2.cox )(1 + 2.cox ) . = − sen x.(1 + 2. cos x ) 1 − 2. cos x 1 − 2. cos x 1 − 2. cos x 1 sen x − cos x sen x − cos x = lim (− cos x ) = − 220. lim =? à lim x →π 4 1 − tgx x →π 4 1 − tgx x →π 4 2 sen x − cos x sen x − cos x sen x − cos x sen x − cos x sen x − cos x f (x ) = = = = = = 1 − tgx sen x sen x cos x − sen x − 1.(sen x − cos x ) 1− 1− cos x cos x cos x cos x sen x − cos x cos x− . = − cos x 1 cos x − sen x21. lim (3 − x ). cos sec(πx ) = ? à lim (3 − x ). cos sec(πx ) = 0.∞ x→3 x→3 3− x 3− x f (x ) = (3 − x ). cos sec(πx) = (3 − x ). 1 1 = = = = sen (πx ) sen (π − πx ) sen (3π − πx ) π . sen (3π − πx ) π .(3 − x ) 1 à lim (3 − x ). cos sec(πx ) = lim 1 1 =π . sen (3π − πx ) x→3 x→3 π . sen (3π − πx ) π (3π − πx ) (3π − πx ) 1 122. lim x. sen( ) = ? à lim x. sen( ) = ∞.0 x→∝ x x→∝ x 1 sen  x sen t 1  x → +∞lim = lim =1 à Fazendo t = x →∝ 1 t →0 t x t → 0 x 4
  5. 5. Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos π 2. sen 2 x + sen x − 1 1 + sen 2. sen 2 x + sen x − 1 1 + sen x 6 =23. lim = ? à lim = lim = x →π 2. sen x − 3. sen x + 1 2 x →π 2. sen 2 x − 3. sen x + 1 x →π − 1 + sen x π 6 6 6 − 1 + sen 6  1 1  sen x − .(sen x + 1) 1+ 2 =−3 à f (x ) = 2. sen x + sen x − 1 2 = 2 = (sen x + 1) = 1 + sen x −1+ 1 2. sen x − 3. sen x + 1  2 1  sen x − .(sen x − 1) (sen x − 1) − 1 + sen x 2  2 πx πx πx24. lim(1 − x ).tg   = ? à lim(1 − x ).tg   = 0.∞ à f (x ) = (1 − x ).tg   =       x →1  2 x →1  2  2  π (1 − x ) = 2 .(1 − x ). π = 2 2 2  π πx  π π (1 − x ). cot g  −  = = à 2 2   π πx   π πx   π πx   π πx  tg  −  tg  −  tg  −  tg  −  2 2  2 2  2 2  2 2  π  π πx  .(1 − x )  −  2 2 2  2 2  πx  π 2lim(1 − x ).tg   = lim = π = Fazendo uma mudança de variável,x →1  2 x →1  π πx  tg (t ) π tg  −  lim  2 2  t →0 t  π πx   −  2 2  π πx  x → 1temos : t= −  2 x t → 0 1− x 2 1− x 2 1+ x 225. lim = ? à lim = lim = x →1 sen (πx ) x →1 sen (πx ) x →1 π . sen (π − πx ) π (π − πx ) f (x ) = 1− x 2 = (1 − x )(1 + x ) = . 1+ x = 1+ x = 1+ x sen πx sen (π − πx ) sen (π − πx ) π . sen (π − πx ) π . sen (π − πx ) (1 − x ) π .(1 − x ) (π − πx ) π  π 26. lim cot g 2 x. cot g  − x  = ? à lim cot g 2 x. cot g  − x  = ∞.0 x →0 2  x →0 2  π  tgx 1 − tg 2 x 1 − tg 2 x f (x ) = cot g 2 x. cot g  − x  = cot g 2 x.tgx = tgx = = tgx. = 2  tg 2 x 2tgx 2.tgx 2 1 − tg 2 x π  1 − tg 2 x 1lim cot g 2 x. cot g  − x  = lim =x →0 2  x→0 2 2 cos x − 3 cos x −t2 127. lim = lim =− x →0 sen 2 x t →1 1 + t + t 2 + ... + t 10 + t 11 12 cos x − 3 cos x t3 − t2 − t 2 .(1 − t ) −t2 f (x ) = (1 − t ).(1 + t + t 2 + ... + t 10 + t 11 ) = = = sen 2 x 1 − t 12 1 + t + t 2 + ... + t 10 + t 11 x → 0t = 2.3 cos x = 6 cos x  t 6 = cos x , t 12 = cos 2 x , sen 2 x = 1 − t 12 t → 1 5
  6. 6. Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidosBriotxRuffini : 1 0 0 ... 0 -1 1 • 1 1 ... 1 1 1 1 1 ... 1 0 sen 2 x − cos 2 x − 1 sen 2 x − cos 2 x − 1 π = lim (− 2. cos x ) = − 2. cos 228. lim = ? à lim = − 2. = π x→ 4 cos x − sen x π x→ 4 cos x − sen x π x→ 4 4 2 − 2 f (x ) = = ( = ) sen 2 x − cos 2 x − 1 2. sen x cos x − 2 cos 2 x − 1 − 1 2. sen x. cos x − 2 cos 2 x + 1 − 1 = cos x − sen x cos x − sen x cos x − sen x2. sen x. cos x − 2 cos 2 x − 2. cos x.(cos x − sen x ) = = −2. cos x cos x − sen x cos x − sen x sen ( x − 1) sen (x − 1) 1 sen (x − 1) 2 x − 1 + 129. lim = ? à lim = lim . . =1 x →1 2x − 1 − 1 x →1 2 x − 1 − 1 x →1 2 (x − 1) 1 sen (x − 1) sen (x − 1) 2x − 1 + 1 sen ( x − 1) 2 x − 1 + 1 sen ( x − 1) 2 x − 1 + 1 f (x ) = = . = . = . = 2x −1 −1 2x − 1 − 1 2x − 1 + 1 2x − 1 − 1 1 2.(x − 1) 11 sen (x − 1) 2 x − 1 + 1 . .2 (x − 1) 1 π − x    sen  3 π + x   2   1 − 2. cos x 1 − 2. cos x    30. lim = ? à lim = lim 2. sen 3 . = π x→ x− π π x→ x− π x →π  2  π − x   3 3 3    3  3 3  2      π +π   2π   2. sen 3 3 . = 2. sen 3 . = 2. sen π . = 2. 3 = 3    2    2    3 2     π + x  π − x       1   π  2.(− 2 ) sen 3 . sen 3 2 . − cos x  2 . cos − cos x   2    2   1 − 2. cos x  =   =     f (x ) = =  2 3 = π π π π − x  x− x − x −   3 3 3 − 1.2. 3  2     π + x  π − x  π − x       2. sen 3 . sen 3 sen 3  2    2   π + x   2       = 2. sen 3 .   2  π  π − x      3     3−x  2     2        1 − cos 2 x 1 − cos 2 x 2. sen x31. lim =? à lim = limπ =2 x →0 x. sen x x →0 x. sen x x x→ 3 6
  7. 7. Limites Trigonométricos Resolvidos Sete páginas e 34 limites resolvidos 1 − cos 2 x 1 − ( − 2 sen x ) 1 − 1 + 2 sen 2 x 2. sen x 2. sen x 2 2 1f (x ) = = = = = x. sen x x. sen x x. sen x x. sen x x x x 1 + sen x + 1 − sen x 1+132. lim = ? à lim = lim = x →0 1 + sen x − 1 − sen x x →0 1 + sen x − 1 − sen x x →0 2. sen x 2.1 x =1f (x ) = x = ( x. 1 + sen x + 1 − sen x ) = x.( 1 + sen x + 1 − sen x )= 1 + sen x − 1 − sen x 1 + sen x − (1 − sen x ) 1 + sen x − 1 + sen x ( x. 1 + sen x + 1 − sen x )= 1 + sen x + 1 − sen x = 1+1 =1 2. sen x sen x 2.1 2. x cos 2 x cos x + sen x 2 233. lim = lim = + = 2 x→0 cos x − sen x x →0 1 2 2 cos 2 x.(cos x + sen x ) cos 2 x.(cos x + sen x ) cos 2 x.(cos x + sen x )f (x ) = cos 2 x = = = = cos x − sen x (cos x − sen x )(cos x + sen x ) . cos 2 x − sen 2 x cos 2 xcos 2 x.(cos x + sen x ) cos x + sen x 2 2 = = + = 2 cos 2 x 1 2 2  3  π 2. − sen x     2  2. sen − sen x  3 − 2. sen x 3 − 2. sen x lim   lim   334. lim =? à lim = = = π π π π π π π π x→ 3 x− x→ 3 x− x→ 3 x− x→ 3 x− 3 3 3 3  π  π    π − 3x   π + 3x    −x  + x       2. sen 3 . cos 3  2. sen 3 . cos 3    2   2    2   2                     lim = lim  = π π π 3x − π x→ 3 x− x→ 3 3 3   π − 3x   π + 3x   2. sen  . cos    6   6  lim x→ π − 1.(π − 3 x ) 3 335. ? 7

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