### 有理函數的圖形 - Proera

```微積分數位化教材
C. L. Lang, Department of Applied Mathematics, I-Shou University
1






C. L. Lang, Department of Applied Mathematics, I-Shou University
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

()
=
()


C. L. Lang, Department of Applied Mathematics, I-Shou University
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



()
=
()
∈ | () ≠ 0
C. L. Lang, Department of Applied Mathematics, I-Shou University
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





2 +  − 2
=
2 − 1
∈ |  ≠ ±1
C. L. Lang, Department of Applied Mathematics, I-Shou University
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





2 +  − 2
=
2 + 1
C. L. Lang, Department of Applied Mathematics, I-Shou University
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2 +  − 2
=
2 − 1


C. L. Lang, Department of Applied Mathematics, I-Shou University
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

1.
2.

C. L. Lang, Department of Applied Mathematics, I-Shou University
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



3 − 1
= 2
−1

3 − 1 ( − 1)( 2 +  + 1)
=
2
−1
( − 1)( + 1)
的定義域為  ∈ |  ≠ ±1 ，其圖形為
C. L. Lang, Department of Applied Mathematics, I-Shou University
9
3 − 1
= 2
−1


C. L. Lang, Department of Applied Mathematics, I-Shou University
10
1.

3 − 1
= 2
−1

2.

C. L. Lang, Department of Applied Mathematics, I-Shou University
11

1.
2.

C. L. Lang, Department of Applied Mathematics, I-Shou University
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



()
=
()

3 2 +  − 1
=
2 − 1
C. L. Lang, Department of Applied Mathematics, I-Shou University
13
3 2 +  − 1
=
2 − 1

C. L. Lang, Department of Applied Mathematics, I-Shou University
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

C. L. Lang, Department of Applied Mathematics, I-Shou University
15


2 +  − 2
=
2 − 1

C. L. Lang, Department of Applied Mathematics, I-Shou University
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2 +  − 2
=
2 − 1


C. L. Lang, Department of Applied Mathematics, I-Shou University
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


3
(1, )
2

C. L. Lang, Department of Applied Mathematics, I-Shou University
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


deg () > deg ()由除法原理可得
=     + ()

()
()
=
=  +
。
()
()

()

()

x 越來越大時有理函數   的圖形越來越接近

C. L. Lang, Department of Applied Mathematics, I-Shou University
19


4 − 2 +  + 2
=
2 − 1
4 − 2 +  + 2
+2
2
= + 2
2
−1
−1
C. L. Lang, Department of Applied Mathematics, I-Shou University
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


C. L. Lang, Department of Applied Mathematics, I-Shou University
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



()
()

()
()

C. L. Lang, Department of Applied Mathematics, I-Shou University
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
= (1  + 1 )1 ⋯ (  +  )
(1  2 + 1  + 1 )1 ⋯ (  2 +   +  )


()
()

=

=1
=1 (  +  )

+
+
2

=1
=1 (  +   +  )
C. L. Lang, Department of Applied Mathematics, I-Shou University
23







2
2 − 1
2 − 1 = ( − 1)( + 1)
2

=
+
2
−1 −1 +1

2 = ( + 1) + ( − 1)
C. L. Lang, Department of Applied Mathematics, I-Shou University
24


==1
2
1
1
=
+
2
−1 −1 +1
C. L. Lang, Department of Applied Mathematics, I-Shou University
25




3 2 − 2 + 5
3 − 1
3 − 1 = ( − 1)( 2 +  + 1)
3 2 − 2 + 5

+
=
+ 2
3
−1
−1  ++1
C. L. Lang, Department of Applied Mathematics, I-Shou University
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

3 2 − 2 + 5
= ( 2 +  + 1) + ( + )( − 1)



3 2 − 2 + 5
2
−3
=
+ 2
3
−1
−1  ++1
C. L. Lang, Department of Applied Mathematics, I-Shou University
27






4
3 − 2 −  + 1
3 −  2 −  + 1 = ( − 1)2 ( + 1)
4

=
+
+
3
2
2
−  −  + 1  − 1 ( − 1)
+1
C. L. Lang, Department of Applied Mathematics, I-Shou University
28



4 = ( 2 − 1) + ( + 1) + ( − 1)2

= −1 ，代入  = 0 得 − + 2 − 1 = 0得  =
1
4
1
2
1
=
+
−
3
2
2
−  −  + 1  − 1 ( − 1)
+1
C. L. Lang, Department of Applied Mathematics, I-Shou University
29


2 4 −  3 + 2 2 + 1
5 −  4 + 2 3 − 2 2 +  − 1

5 −  4 + 2 3 − 2 2 +  − 1 = ( − 1)( 2 + 1)2
 所以
2 4 −  3 + 2 2 + 1
( − 1)( 2 + 1)2

+
+
=
+ 2
− 2
− 1  + 1 ( + 1)2

C. L. Lang, Department of Applied Mathematics, I-Shou University
30

2 4 −  3 + 2 2 + 1
=   2 + 1 2 + ( + )( − 1) + ( + )(
− 1)( 2 + 1)
 比較係數解方程組得
=  = 1,  = −1,  =  = 0
2 4 −  3 + 2 2 + 1
5 −  4 + 2 3 − 2 2 +  − 1
1

=
+ 2
− 2
− 1  + 1 ( + 1)2

C. L. Lang, Department of Applied Mathematics, I-Shou University
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I-Shou University Department of Applied
Mathematics, C. L. Lang
32
C. L. Lang, Department of Applied Mathematics, I-Shou University
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C. L. Lang, Department of Applied Mathematics, I-Shou University
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C. L. Lang, Department of Applied Mathematics, I-Shou University
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C. L. Lang, Department of Applied Mathematics, I-Shou University
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C. L. Lang, Department of Applied Mathematics, I-Shou University
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C. L. Lang, Department of Applied Mathematics, I-Shou University
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