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LIMIT TRIGONOMETRI
X A CO
B
E
X² + Y² = 1
Lihat segitiga OBC
Sin x = BC : OBBC = OB . Sin xBC = 1 . Sin x BC = sin x
cos x = OC : OBOC = OB . cos xOC = 1 . cos x OC = cos x
Luas daerah OCE < Luas segitiga OBC< Luas daerah OAB
Dibagi
Jadi :
1.
2.
Jadi : 3.
4.
5.
6.
7.
8.
5.
6.
Contoh soal: tentukan nilai limit berikut
1.
2.
3.
4.
Jawab
1. 2323
x
xox it
3
3sin.2
3 lim
2
31.
2
3
2. .4tan4tan
limlim
x
xox
x
xox itit
4
11.
4
1
x
xox it
4tan
4.4
1 lim
4
4
3.
4.
.2sin
5tan
2sin
5tan limlim
x
xox
x
xox itit
5
5.
2
2.
x
x.
2
5.
2sin
2.
5
5tanlim
x
x
x
xox it
.2sin
2.
5
5tan.2
5 limlim
x
xox
x
xox itit
2
51.1.
2
5
2
2lim )sin21(1
x
xox it
2
2lim sin2
x
xox it
2lim sin
.2
x
xox it
21.2
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Latihan :
1.
2.
3.
4.
5.
xx
xox it
2tan.32
sin
6 2lim
xx
xxox it
2sin
3sinlim
6
)3sin(3
2lim
xx
xx it
x
xxox it
4cos1
3sin.sinlim
x
xxox it
2
sin1sin1lim
Latihan :
1. 7.
2. 8.
3.
4.
5.
6.
3
0
cos 2 cos 2lim
4 .tanx
x x
x x
2
20
sin 3 sin 3 .coslim
4 tanx
x x x
x x
2
cos(3 6) 1lim
sin( 2). tan(2 4)x
x
x x
2
4
1 sin 2lim
cos cos .sinx
x
x x x
23
1 cos( 3)lim( 2 3) tan(4 12)x
x
x x x
2
4
cos 2lim
cos cos .sinx
x
x x x
0
tan 2lim
tan 2 tan 4x
x
x x
0
sin 2lim
tan 4 tan8x
x
x x
Jawab :
1. =
=
=
=
=
=
=
2
0
2
0
2
0 0
2
0
2
cos 2 (1 cos 2 )lim
4 tan
cos 2 (sin 2 )lim
4 tan
sin 2 1limcos2 lim lim
tan 4
sin 2 1(cos0) 2 lim .1.
2 4
11.(2.1) .
4
14.4
1
x
x
x o x x
x
x x
x x
x x
x x
x xx
x x
x
x
3
0
cos 2 cos 2lim
4 .tanx
x x
x x
Jawab :
2. =
=
=
=
=
=
2
20
2
20
2 2
0 0
2 2
0
sin 3 (1 cos )lim
4 tan
sin 3 (sin )lim
4 tan
sin 3 sin 1lim lim lim
tan 4
sin 3 1(3.lim ).1 .1 .
3 4
1(3.1).
4
3
4
x
x
x o x x
x
x x
x x
x x
x x
x x x
x x x
x
x
2
20
sin 3 sin 3 coslim
4 .tanx
x x x
x x
23
3
3
2
3
2
0
2
0
1 cos( 3)lim( 2 3) tan(4 12)
1 cos( 3)lim( 3)( 1) tan 4( 3)
1 coslim
( 3 1) tan 4
1 (1 2sin )2lim
( 4) tan 4
2sin2lim
( 4) tan 4
sin 122. lim lim lim4 tan
x
x
x
x
y
y y o y o
x
x x x
x
x x x
y
y y y
y
y y y
y
y y y
yy
y y
4y
2
0
2
1 sin( / 2) 1 1 42. .lim .lim2 ( / 2) 0 4 4 tan 4
1 1 12. .1 .12 4 4
1 1 12. . .4 4 41
32
y y o
y y
y y
Misal : x - 3 = y x = y + 3 3 - 3 = y 0 = y
3. Jawab :
2
cos(3 6) 1lim
sin( 2). tan(2 4)x
x
x x
4. Jawab :
2
2
2
0
2
0
cos3( 2) 1lim
sin( 2). tan 2( 2)
cos3 1limsin .tan 2
31 2sin 1
2lim
sin .tan 2
32sin
2lim
sin .tan 2
3sin
22. lim lim limsin tan 2
x
y o
y o
y
y y o y o
x
x x
y
y y
y
y y
y
y y
yy y
y y y
2
2
3sin( )3 1 222. .lim .1. .lim32 2 tan 2( )2
3 12.( .1) .( .1)2 29 1
2. .4 29
4
y o y o
y y
yy
Misal : x + 2 = y - 2 + 2 = y y = 0
2
4
2
4
2 0
0 0
0 2
1 sin 2lim
cos cos .sin
cos 2lim
cos (1 sin )
cos 2(45 )
cos 45 (1 sin 45 )
(cos90 )1 12(1 2)
2 20
1 12(1 2)
2 20
x
x
x
x x x
x
x x
5. Jawab : 6. jawab :
2 2
4
4
4
0 0
0
cos sinlim
cos (cos sin )
(cos sin )(cos sin )lim
cos (cos sin )
(cos sin )lim
cos
cos 45 sin 45
cos 451 12 2
2 212
21 1 2
x
x
x
x x
x x x
x x x x
x x x
x x
x
2
4
cos 2lim
cos cos .sinx
x
x x x
=
=
=
=
=
=
=
=
=
=
=
0
0
0 0
0
tan 2lim
tan 2 tan 4tan 2
lim2
tan 2 tan 4lim lim
2 21tan 4
1 2.lim4
1
1 2.11
3
x
x
x x
x
x
x xxx
x xx x
xx
0
0
0 0
0 0
sin 2lim
tan 4 tan8sin 2
lim2
tan 4 tan8lim lim
2 21
tan 4 tan82.lim 4.lim
4 81
2.1 4.11
2
x
x
x x
x x
x
x xxx
x xx x
x xx x
7. Jawab : 8 . jawab :
