作者LuisSantos ( )
看板trans_math
标题Re: [考古] 请问一题淡江考古计算
时间Sat Jul 12 14:48:26 2008
※ 引述《flygey (努力达成目标)》之铭言：
: 1 1
: Find lim (--------- - ------)
: x->0 -1 2
: xsin x x
: 请问这题怎麽计算
-1
令 y = sin x , 则 x = sin(y)
当 x →0 时 , y→0
1 1
lim (----------- - -----)
x→0 -1 2
(x)(sin x) x
1 1
= lim ------------- - ------------
y→0 (sin(y))(y) (sin(y))^2
sin(y) - y
= lim -----------------
y→0 (y)((sin(y))^2)
cos(y) - 1
= lim ---------------------------
y→0 (sin(y))^2 + (y)(sin(2y))
-sin(y)
= lim -----------------------------------
y→0 sin(2y) + sin(2y) + (2y)(cos(2y))
-sin(y)
= lim ------------------------------
y→0 (2)(sin(2y)) + (2y)(cos(2y))
-cos(y)
= lim ---------------------------------------------
y→0 (4)(cos(2y)) + (2)(cos(2y)) - (4y)(sin(2y))
-1 -1
= ----------- = ---
4 + 2 - 0 6
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