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2 2000 $BG/4pAC?tM}(B II

$B$3$NG/$N4pAC?tM}(B II $B$G$O!"%m%T%?%k$NDjM}$H!V(B$n$ $B

$f(x)$ $B$,L58B5i?t(B

$$f(x)=A_0+A_1(x-a)+A_2(x-a)^2++A_3(x-a)^3+A_4(x-a)^4+\cdots$$ (2)

$B$HE83+$5$l$?$H$9$k!#(B(2) $B$K(B $x=a$ $B$rBeF~$9$k$H(B

$$f(a)=A_0+0+0+0+0+\cdots \ \ \Rightarrow \ \ A_0 = f(a)$$

(2) $B$NN>JU$rHyJ,$9$k$H(B

$$f'(x)=A_1+2A_2(x-a)+3A_3(x-a)^2+4A_4(x-a)^3+\cdots$$

$B$3$l$K(B $x=a$ $B$rBeF~$9$k$H(B

$$f'(a)=A_1+0+0+0+\cdots \ \ \Rightarrow \ \ A_1 = f'(a)$$

$B$b$&0l2sHyJ,$7$F(B

$$f''(x)=2\cdot 1A_2+3\cdot 2A_3(x-a)+4\cdot 3A_4(x-a)^2+\cdots$$

$B$3$l$K(B $x=a$ $B$rBeF~$9$k$H(B

$$f''(a)=2\cdot 1A_2+0+0+\cdots \ \ \Rightarrow \ \ A_2 = \frac{f''(a)}{2\cdot 1}$$

$BF1MM$KHyJ,$7$F(B $x=a$ $B$rBeF~$9$k!#(B

$$\begin{eqnarray*}&& f'''(x)=3\cdot 2\cdot 1 A_3+4\cdot 3\cdot 2 A_4(x-a)+\cdots ... ...ots \ \ \Rightarrow \ \ A_3 = \frac{f'''(a)}{3\cdot 2\cdot 1} \end{eqnarray*}$$

$B$3$l$r7+$jJV$9$3$H$G(B

$$A_n=\frac{f^{(n)}(a)}{n!}$$

$B$,F@$i$l$k!#(B

$B$?$@$7!"$3$l$G$O$"$^$j$KM}O@$,$J$52a$.!"$=$b$=$b(B (2) $B$H(B $BCV$/(B ($BE83+$9$k(B) $B$3$H$N0UL#$,J,$+$i$J$$!#(B

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$B4X?t(B $f(x)$ $B$KBP$7$F(B $g(x)$ $B$,(B

$$\frac{f(x)-g(x)}{x^n}\rightarrow 0 \mbox{\ \ ($x\rightarrow 0$\ $B$N$H$-(B)}$$ (3)

$B$H$J$k$H$-!"(B$g(x)$ $B$O(B $f(x)$ $B$r(B ($x=0$ $B$G(B) $n$ $B$B!"$^$?$O(B $n$ $B $B$H8F$V$3$H$K$9$k!#(B

(3) $B$O(B

$f(x)-g(x)$ $B$O(B $x^n$ $B$h$jB.$/(B 0 $B$K6a$E$/(B

$B$3$H$r0UL#$7!"$9$J$o$A(B

$f(x)-g(x)$ $B$O(B $x^n$ $B$h$j(B 0 $B$K6a$$(B

$B$3$H$K$J$k!#(B ($BCm(B: $B$3$l$r(B $g(x)=f(x)+o(x^n)$ $B$H=q$/$3$H$b$"$k(B)

$BDjM}(B 2

$f$, $g$: $n$ $B2sHyJ,2DG=$G!"(B$f^{(n)}$, $g^{(n)}$ $B$,O"B3$G$"$k$H$-!"(B

$$\mbox{$g$\ $B$,(B $f$\ $B$N(B $n$\ $B<!6a;w(B} \ \ \Leftrightarrow \ \ f^{(k)}(0)=g^{(k)}(0)\ (k=0,1,2,\ldots,n)$$

$B>ZL@(B

$n=2$ $B$H$7$F>ZL@$9$k!#(B

($\Rightarrow$)

$$\frac{f(x)-g(x)}{x^2}\rightarrow 0 \mbox{\ \ ($x\rightarrow 0$\ $B$N$H$-(B)}$$ (4)

$B$H2>Dj$9$k!#(B

1. (4) $B$h$j(B
   
   $$f(x)-g(x)=x^2 \frac{f(x)-g(x)}{x^2}\rightarrow 0\mbox{\ \ ($x\rightarrow 0$\ $B$N$H$-(B)}$$
   
   
   $B$H$J$k$,!"(B$f(x)$, $g(x)$ $B$NO"B3@-$K$h$j:8JU$O(B
   
   $$f(x)-g(x)\rightarrow f(0)-g(0)\mbox{\ \ ($x\rightarrow 0$\ $B$N$H$-(B)}$$
   
   
   $B$H$J$k$N$G!"$h$C$F(B
   

   $$f(0)-g(0)=0$$ (5)

   
   
   $B$H$J$k!#(B
2. (4) $B$h$j(B
   
   $$\frac{f(x)-g(x)}{x}=x \frac{f(x)-g(x)}{x^2}\rightarrow 0\mbox{\ \ ($x\rightarrow 0$\ $B$N$H$-(B)}$$
   
   
   $B$H$J$k$,!"0lJ}$G(B (5) $B$K$h$j$3$N:8JU$OJ,;R$bJ,Jl$b(B 0 $B$K<}B+$9$k(B $BITDj7A$G$"$j!"(B
   
   $$\frac{\{f(x)-g(x)\}'}{(x)'}=\frac{f'(x)-g'(x)}{1}\rightarrow f'(0)-g'(0) \mbox{\ \ ($x\rightarrow 0$\ $B$N$H$-(B)}$$
   
   
   $B$H$J$k$3$H$,(B $f'(x)$, $g'(x)$ $B$NO"B3@-$K$h$j$$$($k!#(B $B$h$C$F!"%m%T%?%k$NDjM}$K$h$j(B
   

   $$f'(0)-g'(0)=0$$ (6)

   
   
   $B$H$J$k!#(B
3. (4) $B$N:8JU$O(B (5) $B$K$h$jITDj7A$G$"$j!"(B
   
   $$\frac{\{f(x)-g(x)\}'}{(x^2)'}=\frac{f'(x)-g'(x)}{2x}$$
   
   
   $B$b!"(B(6) $B$K$h$jITDj7A$H$J$k!#$=$7$F!"(B
   
   $$\frac{\{f'(x)-g'(x)\}'}{(2x)'}=\frac{f''(x)-g''(x)}{2}\righ... ... \frac{f''(0)-g''(0)}{2}\mbox{\ \ ($x\rightarrow 0$\ $B$N$H$-(B)}$$
   
   
   $B$H$J$k$3$H$,!"(B$f''(x)$, $g''(x)$ $B$NO"B3@-$K$h$j$$$(!"$h$C$F(B $B%m%T%?%k$NDjM}$K$h$j(B
   
   $$\frac{f''(0)-g''(0)}{2}=0$$
   
   
   $B$9$J$o$A(B
   

   $$f''(0)-g'((0)=0$$ (7)

   
   
   $B$,@.$jN)$D!#(B

$B0J>e(B 1., 2., 3. $B$N(B (5),(6),(7), $B$K$h$j(B

$$f(0)=g(0), \ f'(0)=g'(0), \ f''(0)=g''(0)$$

$B$H$J$k!#(B

( $\Longleftarrow$)

$B5U$K(B

$$f(0)=g(0), \ f'(0)=g'(0), \ f''(0)=g''(0)$$

$B$,@.$jN)$D$H$9$k$H!"%m%T%?%k$NDjM}$K$h$j(B

$$\lim_{x\rightarrow 0}\frac{f(x)-g(x)}{x^2} =\lim_{x\righta... ...g'(x)}{2x} =\lim_{x\rightarrow 0}\frac{f''(x)-g''(x)}{2} =0$$