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¡]1¡^ ¼Æ¦C a_n =(1+1/n)^n
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e=lim(1+1/n)^n ©Î e=1+1+1/2!+1/3!+¡K.1/n!+....
¡]2¡^ §Q¥Î§¨À½¥i¥H±o¨ì
lim(1+1/n)^n=lim(1+1/x)^x ¦b¦¹x¤£¤@©wn¬°¾ã¼Æ
¡]3¡^ ©w¸qa^x ©Mlogx (¥Ha¬°©³)¤¬¬°¤Ï¨ç¼Æ
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(lnx)'=1/x
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ln (e^x)=x
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(1/e^x)(e^x)'=1
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(e^x)'=e^x
¡]4¡^ ¥Ñì©l©w¸qe=lim(1+1/n)^n «h¹ï©ó©Ò¦³¹ê¼Æx
e^x=lim(1+1/n)^(nx) ©Î e^x=(1+1+1/2!+1/3!+¡K.1/n!+....)^x
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e^x=lim(1+x/n)^n ©Îe^x=(1+x+x^2/2!+¡K+.x^n/n!+......)
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e^x=lim(1+1/n)^(nx)=lim(1+x/n)^n ¥B
e^x=lim(1+1+1/2!+1/3!+¡K.1/n!+..)^x=(1+x+x^2/2!+¡K.x^n/n!+....)
¡i1¡j ln lim(1+1/n)^(nx)
=lim ln(1+1/n)^(nx) ¦]¬°lnx¬°³sÄò¨ç¼Æ©Ò¥Hlim¥i´£¨ì¥~±
=x lim ln (1+1/n)^n
=x ln lim(1+1/n)^n
=x lne
=x
¡i2¡j ln lim(1+x/n)^n
=lim ln(1+x/n)^n
=lim n ln(1+x/n)
¥Ot =1/n
=lim ln(1+tx)/t (ù¥²¹F)
=lim x/(1+tx)
=x
¥Ñ¡i1¡j¡i2¡j©Mlnx¬°¤@¹ï¤@¨ç¼Æ±oe^x=lim(1+1/n)^(nx)=lim(1+x/n)^n
¡i3¡je^x =lim(1+1/n)^(nx)
= lim(1+x/n)^n
=1+1+1/2!+1/3!+¡K.1/n!+....