81 bukti bukti_limit_ apiq
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1
Inovasi Pembelajaran Matematika Kreatif
Pembuktian Limit Terpenting
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a b
c
a < b < c
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a b
c
a < b < c
r = 1
r = 1 jika c = x maka
a < b < c
x cos x < sin x < x
cos x < (sin x)/x < 1
1< (sin x)/x < 1
Jadi, (Sin x)/x = 1
Atau, Sin x = x
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4
Keterangan
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a b
c
a < b < c
r = 1
r = 1
Jika c = x maka
Besar sudut = x/r = x/1 = x
Sinx = b/r = b
Cosx = d/r = dd = cosx
Besar sudut =
x = a/cosx
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a b
c
a < b < c
r = 1
c = x
b = sinx
d = cosx
d = cosx a = x(cosx)
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a b
c a < b < c
r = 1 x(cosx) < sinx < x
d = cosx bagi dengan x,
cosx < (sinx)/x < 1
karena x 0
cos(0) < (sinx)/x < 1
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a b
c r = 1
d = cosx
karena x 0
cos(0) < (sinx)/x < 1
1 < (sinx)/x < 1
teori “apit”
(sinx)/x = 1
sinx = x
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9
Latihan
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Hitunglah
Limit x 0
Sin 3x
7x= … = 3x
7x= 3
7
2x
Sin 5x= … = 2x
5x= 2
5
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Hitunglah
Limit x 0
Sin 3x
Sin 8x= … = 3x
8x= 3
8
Sin 6x
Sin 4x= … = 6x
4x= 3
2
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Hitunglah
Limit x 0
2x + Sin 3x
3x + Sin 8x= … = 5x
11x= 5
11
3x + Sin 6x
2x + Sin 4x= … = 9x
6x= 3
2
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Hitunglah
Limit x 0
2x + tan 5x
3x + Sin 8x= … = 7x
11x= 7
11
3x + tan 2x
2x + tan 5x= … = 5x
7x= 5
7
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Buktikan
Limit x 0
tan x
x= 1
tan x = x
tan xx
= (sin x)/(cosx)
x
= (sin x)/(1)
x
= sin x
x= 1
tan x
x= 1
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15
Inovasi Pembelajaran Matematika Kreatif
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Hitunglah
Limit x 0
tan 2x + tan 5x
Sin 8x - tan 4x= … = 7x
4x= 7
4
Sin 3x - tan 2x
Sin 2x + tan 5x= … = 1x
7x= 1
7
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Hitunglah
Limit x 0
tan 2x . tan 5x
Sin 8x . tan 4x= … = 10x2
32x2= 10
32
Sin 3x . tan 2x
Sin 2x . tan 5x= … = 6x2
10x2= 6
10
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Untuk
Limit x 0
Sin x = x = tan x
Cos x = Cos 0 = 1
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19
Alternatif Bukti
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a
b
c
Luas sektor a < Luas segitiga b < Luas sektor c
r = 1
r = 1
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a
b
c
Luas a < Luas b < Luas cr = 1
r = 1 sudut = x
½ (cos x)2 x < ½ cosx . sin x < ½ .1. x
(cos x)2x < cosx . sin x < x
cos x < (sin x)/x < 1/cosx
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a
b
c
r = 1
r = 1 sudut = x
cos x < (sin x)/x < 1/cosx
cos 0 < (sin x)/x < 1/cos 0
1 < (sin x)/x < 1
(sin x)/x = 1
sin x = x
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23
Test Limit
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Pengujian limit secara formal memanfaatkan delta epsilon.
Tetapi anak-anak lebih suka dan mudah paham menggunakan pengujian dengan kalkulator atau komputer.
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Uji dengan kalkulator,
Limit x 0
Sin 3x
Sin 8x= … = 3x
8x= 3
8
Mis x = 0,001
Sin (0,003)
Sin (0,008)= 5,236
13,96= 0,3750
= 0,3750
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Uji dengan kalkulator,
Limit x 0
2x + Sin 3x
2x + Sin 8x= … = 5x
10x= 5
10
Mis x = 0,001 (rad)
0,002+Sin (0,003)
0,002+Sin (0,008)= 0,00499999
0,00999991= 0,500
= 0,5
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Uji dengan kalkulator,
Limit x 0
2x + tan 4x
2x + Sin 8x= … = 6x
10x= 6
10
Mis x = 0,001 (rad)
0,002+tan (0,004)
0,002+Sin (0,008)= 0,00600000
0,00999991= 0,600
= 0,6