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8.�]$ \mid x\mid\geq 1 $.����
$$ 
2\textup{arctan} x+\textup{arcsin} \frac {2x}{1+x^{2}} = \pi \textup{sgnx}. 
$$
\hbox to 0.7cm {\hfill} {\kai��}.\hbox to 0.2cm {\hfill} \textup{sgnx}$ x $Ū�@signum of $ x $,
��w�q�p�U:\\
$$
\textup{sgnx} = \left\{ \begin{array}{ll}
      -1 &\mbox{�Y $ x < 0 $,} \\
      0  &\mbox{�Y $ x = 0 $,} \\
      1  &\mbox{�Y $ x > 0 $.} \end{array}
\right. 
$$

9.L'H$ \hat{o} $pital�w�z���f���u����:�O
$$ 
f(x) = x^{2}\textup{sin} \frac {1}{x},\hbox to 0.7cm {\hfill} \phi(x) = x,
$$ 
\hbox to 0.7cm {\hfill} 
����
$ \hbox to 3.2cm {\hfill}
\lim_{x \rightarrow 0} \displaystyle \frac {f(x)}{\phi(x)} = \lim_{x \rightarrow 0}x\textup{sin} \frac {1}{x} = 0,
$\\
\hbox to 0.7cm {\hfill} ��$ \frac {f^{'}(x)}{\phi^{'}(x)} $��$ x \rightarrow  0 $�ɵL����.\\
 \hbox to 0.15cm {\hfill}$ \ast $10.�]$ f(x) $��$ g(x) $�����U�C����:\\
\hbox to 0.7cm {\hfill}(a)$ f(x), f^{'}(x), \ldots , f^{(n-1)}(x) , g(x), g^{'}(x), \ldots ,g^{(n-1)}(x) $
���b$ [a,b] $���s��.\\
\hbox to 0.7cm {\hfill}(b)$ f^{(n)}(x) $�M$ g^{(n)}(x) $���b$ (a,b) $���s�b.\\
\hbox to 0.7cm {\hfill}(c)$ [a,b] $����$ n+1 $���I$ x_{0} < x_{1} < \cdots < x_{n} $
��$ f(x_i) = g(x_i),i = 0, 1, 2,\ldots , n. $\\
\hbox to 1.1cm {\hfill} ����$ (a,b) $�������@�I$ \xi $��$ f^{(n)}(\xi) = g^{(n)}(\xi) $.\\
\hbox to 0.7cm {\hfill}\kai{��}.\hbox to 0.2cm {\hfill}$ n=1 $�ɧY��Rolle's theorem.\\
\hbox to 0.2cm {\hfill}$ \ast $11.�]$ f(x),g(x),h(x) $����$ [a,b] $�W���s����,�B���b$ (a,b) $�W�i�L��.
�յ�$ (a,b) $\\ \hbox to 0.7cm {\hfill}�������@��$ \xi $��
$$ \left | \begin{array}{ccc}
f^{'}(\xi)& g^{'}(\xi)& h^{'}(\xi) \\
f(a)& g(a)& h(a) \\
f(b)& g(b)& h(b) \end{array}
   \right | = 0. 
$$
\hbox to 0.7cm {\hfill}\kai{��}.\hbox to 0.2cm{\hfill} $ h(x) = 1 $�ɦ^��\textmd{Cauchy}�����ȩw�z.\\
 \hbox to 0.2cm {\hfill}$ \ast $12.Let $ f(x) $ and $ g(x) $ be continuous functions on $ [a,b] $ with $ g(x) \geq 0 $
for all\\  \hbox to 0.7cm {\hfill}$ x\in [a,b]. $ Prove that there is a number $ \xi $ in $ (a,b) $ such that\\
$$ \int^b_af(x)g(x)\,dx = f(\xi)\int^b_ag(x)\,dx $$
\hbox to 0.7cm {\hfill}\kai{��}.\hbox to 0.2cm{\hfill} �����G�s�@\textit{the weighted mean-value theorem for integrals}�� {\kai �ĤG����\\ \hbox to 0.7cm {\hfill}�w�z}.
$ g(x) = 1 $�ɦ^��n���Ǫ����ȩw�z.
\newpage
\hbox to 0.7cm {\hfill} $ \S $6.��n��. \hfill\break
\hbox to 0.7cm {\hfill} �b���`���ڭ̤��Щw�n���[�����@�����s�w�Y�ޤJ��n��������.
�b�ڭ�\\ �w�q�w�n����,
�ڭ̰��w�Q�n����Ʀ���,
�S���w�n�������j���@�������j.
��\\ �F�����o�ǭ���,
\textmd{Cauchy}�ޤJ�F��n��������. 
�����ڭ̬ݤ@�ӨҤl: 
�ڭ̪��D\\ ����N���$ T $,
\textmd{\textup{arctan}$ T $}���i�H�οn��
$$ \int_0^T\frac {1}{1+x^{2}}\,dx $$
����.�j�a�q�`�S�{��\textmd{arctan$ \infty $}$ = \pi/2 $.
�ҥH�ڭ̧Ʊ���n��
$$ \int_0^\infty\frac {1}{1+x^2}\,dx $$
������$ \pi/2. $\\
\hbox to 0.7cm {\hfill} \textmd{Cauchy}�ޤJ�F\kai{��n��}(improper integral)���[��,
�N�O���F�X�z�a�����o��\\ ���D.
�H�U�ڭ̧�H�e�����L���n���٬�\kai{���`�n��}(proper integral).
�ڭ̭n�w�q\\ ����n���e
$$ \int_a^bf(x)\,dx $$
����,�䤤$ a,b $�N����ƩβŸ�$ \pm\infty, $
�ӥB���$ f(x) $���@�w����.
�H�U�ڭ̥�$ I $��\\ �ܦ���n��,
��w�q�n���H�U�ƴڻ���:\\
\hbox to 0.7cm {\hfill} 1)$ a = -\infty,b $�����,�B����N$ T\in(-\infty,b) $,
�n��
$$ \int_T^bf(x)\,dx $$
�������`�n��.�p�G����
$$ \lim_{T \rightarrow -\infty}\int_T^bf(x)\,dx $$
�s�b�B��������,
���ɧڭ̧�$ I $�w�q���ӷ���.\\
\hbox to 0.7cm {\hfill} 2)$ b = \infty,a $�����,�B����N$ T\in(a,\infty) $,�n��
$$ \int_a^Tf(x)\,dx $$
�������`�n��.�p�G����
$$ \lim_{T \rightarrow \infty}\int_a^Tf(x)\,dx $$
�s�b�B��������,
���ɧڭ̧�$ I $�w�q���ӷ���.
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