作者Cayley (水色天藍)
看板trans_math
標題Re: [微分] 請教一題極限
時間Sat Oct 30 10:35:24 2004
※ 引述《Acrylates (金色狂風(別倒著唸))》之銘言：
: lim {√(x^2 - x + 1) - ax - b} = 0 則a,b=? (a,b是實數)
: x→∞
1/x--->0 as x--->infinity
Assume y=1/x and substitute f(x)=sqrt(x^2-x+1)
then we get f(y) = sqrt(y^2-y+1) / y
The Taylor series of sqrt(y^2-y+1) at y=0 is
1 - 1/2 y + 3/8 y^2 + o(y^2)
then f(y) = 1/y - 1/2 + 3/8 y + o(y)
x--->infinity , that is y--->0 ,then 3/8 y ---> 0
o(y) ---> 0
so a = 1 and b = - 1/2
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