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�� 14.1

�Ȥ�� ο� $\epsilon$ �ȼ¿�

��Ϳ��줿�Ȥ��Ȥ��

$\displaystyle \vert x-a\vert<\delta \implies \vert f(x)-f(a)\vert<\epsilon$

��ߤ�� $\delta>0$ ��ĵ󤲡��ºݤˤ�� $\delta$ �� 嵭��Ȥ򼨤��ʤ��

	$\displaystyle \vert f(x)-f(a)\vert$
$\displaystyle =$	$\displaystyle \vert(a+h)^3+3(a+h)^2-5(a+h)+7-(a^3+3 a -5 a +7)\vert$
$\displaystyle =$	$\displaystyle \vert 3 a ^2 h + 3 a h^2 + h^3 +6 a h + 3 h^2 -5 h\vert$
$% latex2html id marker 773 $\displaystyle \leq$$	$% latex2html id marker 774 $\displaystyle \vert 3 a ^2 h\vert +\vert 3 a h^2\vert + \vert h^3\vert +\vert 6 a h\vert + \vert 3 h^2\vert + \vert 5 h\vert\qquad$$
$% latex2html id marker 775 $\displaystyle \underset{\text{(��)}}{\leq}$$	$% latex2html id marker 776 $\displaystyle \vert 3 a ^2\vert \vert h\vert +\vert... ...ert h\vert +\vert 6 a\vert \vert h\vert + 3 \vert h\vert + 5 \vert h\vert\qquad$$
$\displaystyle =$	$\displaystyle (3 \vert a\vert^2+9\vert a\vert+9 ) \vert h\vert \underset{\text{(��)}}{<}\epsilon$

(ȯŸ) ��

�� $\mbox{${\mathbb{R}}$}$ �ǰ��Ϣ³�ǤϤʤ��Ȥ򼨤��ʤ��

�� 14.2 $f(x)=\frac{1}{x^2}$ �Ȥ�� $\epsilon>0$ ��Ϳ��줿�Ȥ��Ȥ��

$\displaystyle \vert x-10\vert<\delta \ \implies\ \ \vert f(x)-f(10)\vert<\epsilon$

��ߤ�� $\delta>0$ ��ĵ󤲡��ºݤˤ�� $\delta$ �� 嵭��Ȥ򼨤��ʤ��

��ʤꤿ�ġ��դ�� $\vert f(10+h)-f(10)\vert$ �򲼤Τ褦��ɾ��ɤ��

	$\displaystyle \vert f(10+h)-f(10)\vert$
$\displaystyle =$	$\displaystyle \left\vert\frac{1}{(10+h)^2}-\frac{1}{10^2}\right\vert$
$\displaystyle =$	$\displaystyle \left\vert\frac{10^2-(10+h)^2}{10^2 (10+h)^2}\right\vert$
$\displaystyle =$	$\displaystyle \left\vert\frac{-20 h - h^2}{10^2 (10+h)^2}\right\vert$
$\displaystyle =$	$\displaystyle \left\vert\frac{1}{10^2 (10+h)^2}\right\vert\cdot \vert 20 h+h^2\vert$
$% latex2html id marker 821 $\displaystyle \leq$$	$% latex2html id marker 822 $\displaystyle \left\vert\frac{1}{10^2 (10+h)^2}\right\vert\cdot( \vert 20 h\vert+\vert h^2\vert) \qquad($$ �� $\displaystyle )$
$% latex2html id marker 824 $\displaystyle \underset{\text{(��)}}{\leq}$$	$\displaystyle \left\vert\frac{1}{10^2 (10+h)^2}\right\vert\cdot( 20 \vert h\vert+\vert h\vert)$
$\displaystyle =$	$\displaystyle \left\vert\frac{1}{10^2 (10+h)^2}\right\vert\cdot( 21 \vert h\vert)$
$% latex2html id marker 828 $\displaystyle \underset{\text{(��)}}\leq$$	$% latex2html id marker 829 $\displaystyle \frac{1}{10^2\cdot 9^2} \cdot 21 \vert h\vert \leq \vert h\vert \underset{\text{(��)}}{<}\epsilon$$

	$\displaystyle \vert h\vert<1,$	(��)
	$\displaystyle (3 \vert a\vert^2+9\vert a\vert+9)\vert h\vert<\epsilon.$	(��)

	$\displaystyle \vert h\vert<1,$	(��)
	$\displaystyle \vert h\vert<\epsilon.$	(��)

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