Pertemuan 3i
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Limit Fungsi Jika x ∞
Oleh DEDEH HODIYAH
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Menghitung Limit Fungsi Trigonometri
Rumus-rumus sbb:
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14
Kekontinuan Fungsi
Fungsi f(x) dikatakan kontinu pada suatu titik x = a jika(i) f(a) ada
ada)(lim xfax(ii)
(iii) )()(lim afxfax
Jika paling kurang salah satu syarat diatas tidak dipenuhi maka f dikatakantidak kontinu di x=a
a
(i)º f(a) tidak ada
f tidak kontinu di x=a
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a
(ii)
1L2L
Karena limit kiri(L1) tidaksama dengan limit kanan(L2)maka f(x) tidak mempunyai limitdi x=a
Fungsi f(x) tidak kontinu di x=a
(iii)
a
●
º
f(a) f(a) ada)(lim xf
axL ada
Tapi nilai fungsi tidak sama denganlimit fungsi
Fungsi f(x) tidak kontinu di x=a
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(iv)
a
f(a)
f(a) ada)(lim xf
axada
)()(lim afxfax
f(x) kontinu di x=2
Ketakkontinuan terhapusKetakkontinuan kasus (i) bisa dihapusdengan cara mendefinisikan nilai fungsidititik tersebut = limit fungsia
º
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contohPeriksa apakah fungsi berikut kontinu di x=2, jika tidak sebutkanalasannya
24)(
2
xxxf
2,3
2,24
)(2
x
xxx
xfa. b.
2,12,1
)( 2 xxxx
xfc.
Jawab :a. Fungsi tidak terdefinisi di x=2 (bentuk 0/0) f(x) tidak kontinu
di x=2b. - f(2) = 3
42lim)2()2)(2(lim
24lim
22
2
2
x
xxx
xx
xxx
)2()(lim2
fxfx
--
Karena limit tidak sama dengan nilai fungsi, maka f(x) tidakkontinu di x=2
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18
c. 312)2( 2 f-
- 31lim)(lim22
xxf
xx
31lim)(lim 2
22
xxf
xx
3)(lim2
xfx
)2()(lim2
fxfx
-
Karena semua syarat dipenuhi f(x) kontinu di x=2
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19
Kontinu kiri dan kontinu kananFungsi f(x) disebut kontinu kiri di x=a jika
)()(lim afxfax
Fungsi f(x) disebut kontinu kanan di x=a jika
)()(lim afxfax
Fungsi f(x) kontinu di x=a jika kontinu kiri dan kontinu kanan di x=aContoh : Tentukan konstanta a agar fungsi
2,12,
)( 2 xaxxax
xf
Kontinu di x=2
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Jawab :Agar f(x) kontinu di x=2, haruslah
f kontinu kiri di x=2)2()(lim
2fxf
x
aaxxfxx
2lim)(lim22
1412)2( 2 aaf
2 + a = 4a – 1 -3a = -3 a = 1
f kontinu kanan di x=2
)2()(lim2
fxfx
1412)2( 2 aaf141lim)(lim 2
22
aaxxf
xx
Selalu dipenuhi
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21
1. Diketahui
1,221,1
)(2
xxxx
xf
selidiki kekontinuan fungsi f(x) di x = -1
Soal Latihan
2. Agar fungsi
2,321,
1,1)(
xxxbaxxx
xf
kontinu pada R, maka berapakah a + 2b ?
3. Tentukan a dan b agar fungsi
2,42
2,24
)(2
xx
xxbxax
xf
kontinu di x = 2
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SekianTerima Kasih