\documentclass[a4j, 12pt]{jarticle}

\usepackage{amsmath}
\usepackage{amssymb}

%---------------------Page Format----------------------
\oddsidemargin=0truecm
\textwidth=16truecm
%---------------------------------------------------------------
\def\l{\lambda}
\def\e{\varepsilon}
\def\d{\delta}

\newcommand{\al}{{\alpha}}
\newcommand{\bt}{{\beta}}

\newcommand{\C}{{\mathbb C}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\N}{{\mathbb N}}

\begin{document}
\begin{center}
{\large ���wI���K�𓚗�@��U��@2007�N6��19���o�蕪} 

\smallskip

�S�� ���n����CTA �O�p �~
\end{center}

\bigskip

\noindent{\bf [1]}
(i) 
$e^x =1+x+\frac{1}{2}x^2+\frac{1}{6}x^3
+O(x^4)$,
$\sqrt{1-x}=1-\frac{1}{2}x-\frac{1}{8}x^2-\frac{1}{16}x^3+O(x^4)
$���A
\[
e^x\sqrt{1-x}=1+\frac{1}{2}x-\frac{1}{8}x^2-
\frac{13}{48}x^3+O(x^4).
\]

\smallskip

(ii) 
$\sqrt{1+x}=1+\frac{1}{2}x-\frac{1}{8}x^2+O(x^3)$,
$\cos{x} =1-\frac{1}{2}x^2+\frac{1}{24}x^4+O(x^6) $ ���A
\begin{eqnarray*}
\sqrt{\cos{x}} = \sqrt{1+(\cos{x}-1)}
= 1-\frac{1}{4}x^2-\frac{1}{96}x^4+O(x^6). 
\end{eqnarray*}

 
\bigskip

\noindent{\bf [2]}  
$\log (1+t)=t-\frac{1}{2} t^2 +\frac{1}{3} t^3
-\frac{1}{4} t^4 +\frac{1}{5} t^5 +O(t^6)$��
$t=x^2 , -x^2$��
����������̂��l���Čv�Z���鎖�ɂ��A
\[
f(x)=\log (1+x^2) -\log (1-x^2)
=2x^2 +\frac{2}{3} x^6 +\frac{2}{5} x^{10} + O(x^{12}).
\]
$x^{10}$�̌W�������ɒ��ڂ��āA
$\displaystyle f^{(10)}(0)=10! 
\frac{2}{5}$
��������B

\bigskip

\noindent{\bf [3]}
�C�ӂ�$x>1$�ɑ΂��āA���ϒl�̒藝���
$\displaystyle \frac{f(x)-f(1)}{x-1} =f'(t)$
���݂����悤��$t\in (1,x)$
�����݂���B���̂Ƃ�
\[
|f(x)|+|f(1)|\ge |f(x)-f(1)|=|f'(t)(x-1)|\ge c|x-1|
\to\infty\quad(x\to\infty)  
\]
���A$\lim_{x\to\infty} |f(x)|=\infty$��������B

\bigskip

���܁A�u�C�ӂ�$M>0$�ɑ΂��āA$f(u)$��$f(v)$���������ł���悤��$u,v>M$
�����݂���v�Ɖ��肷��ƁA
$f$�͘A���֐�������A���Ԓl�̒藝���$u,v$�Ԃ̓_$w$��$f(w)=0$�ƂȂ�悤
�Ȃ��̂����݂��邪�A�����
$\lim_{x\to\infty} |f(x)|=\infty$�ɖ�������B
����Ă���$M>0$�����݂��āA$f(x)$��$x>M$�ɂ����ď�ɓ������B�]����
$\lim_{x\to\infty} f(x)=\infty$�܂���$\lim_{x\to\infty} f(x)=-\infty$
�����藧�B

\bigskip

\bigskip

\noindent{\bf [4]}
$\lim_{x\to c} f'(x)=\al$�Ƃ���B
�C�ӂ�$t\in (a,c)\cup (c,b)$�ɑ΂��āA���ϒl�̒藝���
$c$, $t$�Ԃɓ_$u=u(t)$�����݂���
$\displaystyle 
\frac{f(t)-f(c)}{t-c} =f'(u)$���݂����B
$t\to c$�̂Ƃ�$u\to c$������A
$\displaystyle \lim_{t\to c} \frac{f(t)-f(c)}{t-c} =
\lim_{u\to c} f'(u)=\al$�B
����āA$f$��$c$�ɂ����Ă������\��$f'(c)=\al$�ƂȂ�B

\bigskip

\bigskip

\noindent�E[2]�͑���Taylor�W�J��p�����v�Z���@�ł�
�ł��܂��B���ڌv�Z�͂��Ȃ��悤�ɂ��܂��傤�B 

\bigskip

\noindent�E 
[3]�ł́A�O����݂��悤��$f'(x)$���A��
�֐��ł���ۏ؂͂���܂���B
�]���āA$|f'(x)|\ge c$�̉��肩�炽������
�u$\forall x$, $f'(x)\ge c$�v�܂���
�u$\forall x$, $f'(x)\le -c$�v�Ƃ͂Ȃ�Ȃ����ɒ��ӂ��K�v�ł��B

�܂��A
$\lim_{x\to\infty} |f(x)|=\infty$��
�������Ƃ��āA���ꂾ���ł͂܂����_��
�����ꂽ���ɂ͂Ȃ�܂���B�𓚗�̌㔼�ł́A$f$��$x\to\infty$�̂Ƃ���
�u�U�����Ȃ��v����
$f(x)$�̘A�����ɒ��ڂ��ďؖ����Ă��܂��B

\bigskip

\begin{flushright}(�𓚗�쐬�F�O�p)
\end{flushright}

\end{document}