KALDIF 2.6 TEOREMA LIMIT UTAMA
Kalkulus Differensial – Nur Insani 2012
2.6. TEOREMA LIMIT UTAMA
Andaikan bil. bulat positif, konstanta, & fungsifungsi yg mempunyai limit di �. Maka:
�
1.
=
→�
�
2.
=
→�
�
�
3.
=
→�
→�
�
�
�
4.
±
=
±
→�
→�
→�
�
�
�
5.
.
=
.
→�
→�
→�
�
→�
6.
7.
8.
�
→�
�
→�
asal
Contoh:
1.
=
=
=
�
→�
�
3
→ −1
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�
→�
�
→�
�
→�
, asal
�
→�
�
→�
≠0
,
> 0 jk n genap.
2
−1=
�
3
→ −1
2
−
�
1
→ −1
1
Kalkulus Differensial – Nur Insani 2012
�
2
=3
−1=3
→ −1
= 3 −1 2 − 1 = 2
2.
=
4
�
→ −2
4
3
�
→ −2
+6 =
3
�
→ −2
4
�
6 =
→ −2
+
2
�
→ −1
3
4
−1
+6
−8 − 12
−20 (tidak ada).
�
∴
harus > 0 jk n genap (sifat 8).
→�
=
4
Teorema Substitusi
Jika suatu
fungsi polinom atau fungsi rasional,
maka
�
= �
→�
asalkan dlm kasus fs. rasional, penyebut ≠ 0.
Contoh:
�
1.
→1
3
+2
= 1
3
2
+
+6
+2 1
2
+ 1 + 6 = 10
2
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Kalkulus Differensial – Nur Insani 2012
Soal Tambahan
5 3 +8 2
�
1.
→ 0 3 4 −16 2
3 −1
�
2.
→ 1 −1
�
−4
3.
→ 4 −2
4
1
�
− 2
4.
−4
→ 2 −2
�
5.
→ 0−
�
6.
→ −2− +2
3
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2.6. TEOREMA LIMIT UTAMA
Andaikan bil. bulat positif, konstanta, & fungsifungsi yg mempunyai limit di �. Maka:
�
1.
=
→�
�
2.
=
→�
�
�
3.
=
→�
→�
�
�
�
4.
±
=
±
→�
→�
→�
�
�
�
5.
.
=
.
→�
→�
→�
�
→�
6.
7.
8.
�
→�
�
→�
asal
Contoh:
1.
=
=
=
�
→�
�
3
→ −1
[email protected]
�
→�
�
→�
�
→�
, asal
�
→�
�
→�
≠0
,
> 0 jk n genap.
2
−1=
�
3
→ −1
2
−
�
1
→ −1
1
Kalkulus Differensial – Nur Insani 2012
�
2
=3
−1=3
→ −1
= 3 −1 2 − 1 = 2
2.
=
4
�
→ −2
4
3
�
→ −2
+6 =
3
�
→ −2
4
�
6 =
→ −2
+
2
�
→ −1
3
4
−1
+6
−8 − 12
−20 (tidak ada).
�
∴
harus > 0 jk n genap (sifat 8).
→�
=
4
Teorema Substitusi
Jika suatu
fungsi polinom atau fungsi rasional,
maka
�
= �
→�
asalkan dlm kasus fs. rasional, penyebut ≠ 0.
Contoh:
�
1.
→1
3
+2
= 1
3
2
+
+6
+2 1
2
+ 1 + 6 = 10
2
[email protected]
Kalkulus Differensial – Nur Insani 2012
Soal Tambahan
5 3 +8 2
�
1.
→ 0 3 4 −16 2
3 −1
�
2.
→ 1 −1
�
−4
3.
→ 4 −2
4
1
�
− 2
4.
−4
→ 2 −2
�
5.
→ 0−
�
6.
→ −2− +2
3
[email protected]