³o¬O¥Ø«e¥þ¤å,¼È®É¦³ÂIõì,¤é«á¥i¯à·|¦³¨Çקï:
----------------------------
¦¹¤å¥Øªº¦b©ó«Ø¥ß¹q¸£ºâªk°ò¦
¦¹Àɸm©ó
https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-zh.txt/download
¨Ã·|¤£®É§ó·s.
+------+
| ¹ê¼Æ |
+------+
n-¶i¨î©w㸃¼Æ::= ¥Ñn-¶i¨î¼Æ¦r¦ê©Òªí¹Fªº¼Æ. ¦¹¦r¦ê¥i§t¤@Ót¸¹©Î¤p¼ÆÂI:
<fixed_point_number>::= [-] <dstr1> [ . <dstr2> ]
<dstr1>::= 0 | <nzd> { 0, <nzd> }
<dstr2>::= { 0, <nzd> } <nzd>
<nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9) // '¼Æ¦r'·|ÀH¶i¨î§ïÅÜ
Ä´¦p: 78, -12.345, 3.1414159...(£k)
¹ï©ó¥ô¨â¬Û¦Pn-¶i¨îªº¶i¨î¼Æx,y, x,y©Òªíªº¼Æ¬Û¦P iff x,yªº
<fixed_point_number>ªí¹F§¹¥þ¬Û¦P.
¹ê¼Æℝ::= {x| x爲n-¶i¨î©w㸃¼Æ²Å¸¹©Ò¯àªí¹F¤§¼Æ. ªí¹Fxªº¼Æ¦r²Å¸¹¥iµLªø(°Ñªþ䤸)}.
¹ê¼Æ©w¸q¥²¶·¯à´£¨Ñ¤@ºØºtºâªk(®e³\µLºë«×)¨Ó¼Ð¥Üª«²z¤Wªº¨è«×¤Ø. ´N³oÂI
¨Ó»¡,¹ê¼Æ¨Ã¤£§¹¥þ¬O¯Â²z½×,¤]¬O¯à¥Î¼Æ¾Ç´yz¦t©zªº°ò¥»ì¦].
µù: µLªø(µLªk¦³ªí¹F)ªº¼ÆµLªk¥þ³£¦³©ú½T©w¸q,¦]爲¦s¦³¤£¥iªí¹Fªº¼Æ.
³o¬O¥HÂ÷´²²Å¸¹ªí¹F¹ê¼Æªº¤@Ó¯S©Ê(¦ü¶q¤l?). Ä´¦p:
A= lim(n->¡Û) 1-3/10^n = lim 0.999... =1
B= lim(n->¡Û) 1-2/2^n = lim 0.999... =1
C= lim(n->¡Û) 1-1/n = lim 0.999... =1
...
¥tºØ»¡ªk¬O 10*0.999... ³sÄò¼10¥u¯à¬Ý¨ì«e±ªº9,¬Ý¤£¨ì§À¬qªº¼Æµ²ºc.
(10*0.999... ªº³B²z¤è¦¡§ïÅܤF¼Æªºµ²ºc. ¹ï©óµL¯Å¼Æ,³oÂI«Ü«n,
¦]µL¯Å¼Æ©Ò©w¸qªº¼Æ¥i¯à¦]¦¹¤v§ïÅÜ)
¥Ñ©ó<fixed_point_number> «D±`©ú½Tªº¯u¹ê¥B²o§è¨ìµL,¥ô¦³¦r²Õ¦¨ªº
²z½×µLªk§¹¥þ´yzℝ,'§¹³Æ'¬O¤£¥i¯àªº.
µù: ¦¹©w¸q隠§tªí©ú´`Àô¤p¼Æ爲µL²z¼Æ. ¶¶«K²µu»¡©ú±`¨£ªº¥N¼ÆÅ]³Nµý©ú:
(1) x= 0.999...
(2) 10x= 9+x // 10x= 9.999...
(3) 9x=9
(4) x=1
¸Ñµª: ¨S¦³¤½²z©Î©w²z¥iµý(1) => (2)
µù: §PÂ_´`Àô¤p¼Æx¬O§_爲¦³²z¼Æ¥i¥H³sÄòªº´î¥h´`Àô³¡¥÷p(i). ¦pªG¦b¦³¨BÆJ
¡@¡@¡@¡@¡@ ¤º x-p(1)-p(2)-...=0, «hx爲¦³²z¼Æ,§_«h爲µL²z¼Æ. ¦]爲Yx爲¦³²z¼Æ,
³Ñ¾l³¡¥÷ r(i)= x-p(1)-p(2)-... ¥²¶·爲¤@´`Àô³¡¥÷p(i). ¦ý®Ú¾Ú¡¥´`Àô¡¦
©w¸q, r(i)¤£爲p(i). ¦]¦¹,´`Àô¤p¼Æ爲µL²z¼Æ.
¹ê¼Æ°ò¥»¤W´N³o麽²³æ. ·¥¬O¥Î¨Ó©w¸q·L¤À,¤Î´£¨Ñ´M§ä¾É¼Æªº¤èªk,»P¹ê¼Æ©w¸qµLÃö
(Y¦³Ãö«Y,Ãþ¦ü¥H¤Wªº'ℝ'¤]n¥ý©w¸q,§_«h«ÜÃøÁ׶}¹PÀô½×µý°ÝÃD).
+------+
| ·¥ |
+------+
·¥::= lim(x->a) f(x)=L
http://www.math.ntu.edu.tw/~mathcal/download/precal/PPT/Chapter%2002_04.pdf
http://www.math.ncu.edu.tw/~yu/ecocal98/boards/lec6_ec_98.pdf
https://en.wikipedia.org/wiki/Limit_(mathematics)
https://en.wikipedia.org/wiki/Limit_of_a_function
https://www.geneseo.edu/~aguilar/public/notes/Real-Analysis-HTML/ch4-limits.html
«ÂI: L爲º¡¨¬£`-£_±Ôz±¡ªp¤U,xÁͪña®Éªº"·¥È"(f(a)¥i¥H¥¼©w¸q,Áöf©óa³sÄò®É,
L=f(a)). L爲¨Ì¦¹"©w¸q"¥XªºÈ,¤£¬O"µLÁͪñ...§Y¬Ûµ¥"(¨S¦³³oºØÅÞ¿è).
Ä´¦p1: A= lim(n->¡Û) 1-1/n= lim(n->0⁺) 1-n= lim 0.999...=1
B= lim(n->¡Û) 1+1/n= lim(n->0⁺) 1+n= lim 1.000..?=1
Ä´¦p2: A=lim(x->ℵ₀) f(x), B=lim(x->ℵ₁) f(x) // ℵ₀,ℵ₁¬O§_«ê·í¬O¥tÓ°ÝÃD,¦ýY
¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@// ±Ä¡§³Ì²×±N¬Û¦P¡¨¸ÑÄÀ,·|¦³°ÝÃD
¡§·¥¡¨©w¸qA=B, ¦ý¤£¬O¤º®e¬Ûµ¥. Y±Ä"xÁͪñ..«hµ¥©ó.."»¡ªk, «h¦³«Ü¦hÅÞ¿è¤Wªº
°ÝÃD.
µù:·¥¬O©w¸q¦b¤v¦sªº¼Æ¨t¤W,µLªk¥Î¨Ó©w¸q¨ä©ÒÁͪñªº¼Æ.
µù:·¥ªº¼ªk¤½¦¡(lim(x->c) (f(x)*g(x))= (lim(x->c) f(x))*(lim(x->c) g(x)) )
¥i¯à¦³ÂI°ÝÃD:
³]A=lim(n->¡Û) (1-1/n)= 1
A*A*..*A= ... = lim(n->¡Û) (1-1/n)^n // 1=1/e ?
+------------------+
| (ÁÙì)·L¿n¤À¸àÄÀ |
+------------------+
http://www.math.ntu.edu.tw/~mathcal/download/precal/PPT/Chapter%2002_08.pdf
°²³]·L¿n¤À¬O¥Ñ¨Dºâ¨ç¼Æ±¿n°ÝÃD¶}©l: ³]F爲fªº±¿n¨ç¼Æ. ¥Ñ¨ç¼Æ±¿n¸q·N¥i±o:
(F(x+h)-F(x)) ¡Ü (f(x+h)+f(x))*(h/2) // h¬OÓ¨¬°÷¤pªº°¾²¾(´ú¸Õ)¶q
<F>0)¬Of(x)
F´Á±æ©Ê½è: (1)»~®t|lhs-rhs|ÀH·L¤p°¾²¾¶qhÄY®æ»¼´î(°Ñµù1) (2)h=0®É,lhs=rhs
¦]lhsªºh¤£¯à¬O0,©Ò¥H·L¿n¤Àªº°ò¥»°ÝÃD´N¬O´M¨D¯à¨Ï¥H¤W±Ôz¦¨¥ßªºF(©Îf)...
¬G,¤W±Ôz¥iªí¹F¦¨:
D(f(x))= lim(h->0) (F(x+h)-F(x))/h = f(x)
µù1: ½Ò¥»ªº»¡ªk¦³ÂI¤£¦P,¦ýF´Á±æ©Ê½èn¨D©Ò©w¸qªº¼Æ(§Y·¥ÈL)¶·°ß¤@.
µù2: §Æ±æ³oºØ¸ÑÄÀ¥iÁ׶}¤£¥²nªºµL¤j(©Î¤p)ªº¸ÑÄÀ°ÝÃD,¶i¦Ó´£¨Ñ¤@¨Ç®¯½×©Î²z½×
¸û¥¿½Tªº¸ÑÄÀ°ò¦,¦pZeno®¯½×,´`Àô¤p¼Æ,...,¨ä¹êÁÙ¦³§ó¦h(«ü¼Æ,Cantor¶°¦X,
µL¯Å¼Æ...).
+------+
| ªþ¿ý |
+------+
ªþ¿ý1: ℝ,ℕ¶°¦X¤£¥i1-1¹ïÀ³.
ÃÒ: ³]X= {x| x爲Peano¤½²z©Ò©w¸qªº¼Æ¤Î¥]§tµ{§Ç¤£²×¤î©Ò¯àªíªº¼Æ(¼Æ©Ê¥Ñ«áÄ~²£¥Í
¡@¡@¾¹©w¸q}, «hX»Pℕªº¤¸¯À¶¡µLªk«Ø¥ß1-1¹ïÀ³ºâªk(³æ¯Â¦]爲²×¤î/¤£²×¤îì¦]). ¤S¦]X
»Pℝ ¦Pºc, ¬Gℝ,ℕ¶°¦X¤£¥i1-1¹ïÀ³ (¦¹µý¦³ÂI°¨ªê,À³¸Ó¨S°ÝÃD).
ªþ¿ý2: §Ú¹ïµL¤j¡Ûªº¸gÅ礣¨¬. ³oùØ¥u´£¨Ñ¨Ç¸û½T©wªºì«h:
1.µL¤j¡Û¤]¥]§t¦bℝ¤¤,µL¤j»P¤@¯ë¦³¼Æ¬Û¦P,¯à¹Bºâ,¯à¤ñ¤j¤p,¥u¬O¥¦¦³µL¦hÓ.
Á|¤@±`¨£¥Ù¬Þ¨Ò¤l: "1+¡Û =¡Û"»P"lim(x->¡Û) f(x)"ªºÁͪñ·§©À¥Ù¬Þ. ¦]Y¬O,«h¤À¤£¥X
¡@¡@¬O§_x+1,©Îx-1¨ºÓ¬OÁͪñ©Î»·Â÷¡Û. '¡Û'ªº»y·N¥²¶·©T©w.
µù:Y¹ê¼Æ¤£§tµL¤j,«h'¹L®Éªº¹ê¼Æ'¤£À³¦³µL¯Å¼Æ. '¹L®Éªº¹ê¼Æ'¤£¥i¯à¦P®Éµý©ú
爲¤@P»P§¹³Æ. Ä´¦p¤@¨Ç¥H¼Æ©w¸q¼Æªº¹ê¼Æ©w¸q(¤£¦¨¥ß) («Ü©êºp,¥»¤H¤£²M·¡¹ê»Ú
¤Wªº'¹L®Éªº¹ê¼Æ'¦³¦h¤ÖÓ). Y¤@P,«h³Ì¦h,¥¦¬O¦¹ÀÉ©w¸qªº¹ê¼Æªº¤l¶°.
2.µL¤j¡Ûªº»y·N°ò¥»¤W¬O«ü¤@Óµ{§Ç°j°é(¦p:Peano¤½²z). ¯S©Ê¬O"¥Ã¤£²×¤î" (¦¹©ó
µ{§Çªºµý©ú¤¤¥i¯à·N«ä¬O'¤£¥i¨M©w'©Î¡¥¥Ù¬Þ¡¥). ¥tÓ¬ÛÃö°ÝÃD¬O¸Y±K©Êªº±Ôz¤£§¹¾ã,
¦]¹ï©ó¦³²z¼Æ,¸Y±K©ÊªºÀ³¥Î¥²¶·²×¤î,¹ê¼Æ«hµL¦¹n¨D.
3.µL¤j¡Û¤]¥]§tCantor§Ç¼Æ,¦]爲'¼Æ'ªº·§©À«Üì©l. ÁöµM'§Ç¼Æ'¦³¨ä©w¸q,¥¦¤]¥Î¨ì¤F
¡@¡@p¼Æ·§©À(¥t,ℝ¤¤ªº¥[¼¹Bºâ¬O«Ê³¬ªº).
ªþ¿ý3: ¤Gºû¼Æ¥iªí¥±. ¦b¤Gºû¼ÆªºÆ[ÂI¤U,¥unº¡¨¬¶ZÂ÷¤½³] (1.ÂI¸s¥²¾¤£§ïÅÜÂI¸s
¶¡ªº¶ZÂ÷ 2.ÂI¸sªº«Y¼Æ¿n¤£§ïÅÜÂI¸s¶¡ªº¶ZÂ÷¤ñ) §Y¥i«Øºc¼Ú¤ó´X¦ó¨t²Î.
³oùØ·Q»¡ªº¬O: ³oºØ'½èÂI¦t©z'¬O®Ú¾Ú§Ú̪º¹w³]©Ê½è©Ò«Øºcªº. §Ú̲רs¬O¦b±´¯Á
ª¾ÃѦۨªº»y·N. ¨Ã¥B,¥unÅ޿覨¥ß,À³³£¥i§ä¨ì¬ÛÀ³¨Æ¹ê. ¤Ï¦V¨Ó»¡,¥Ñª«²z¨Ó±´¯Á
'¹ê¼Æ'°ò¥»¤W¬O¦¨¥ßªº. ¨Ì¼Æ¦ì®É¥Nªº»¡ªk,¦t©z¬OÓpºâ¾÷.
------------------------------------------------------------------------------