Re: [中學] 一題極限問題請教~
: 不知道該如何消掉零因子
: 想請教大師指點方向，感謝您，謝謝。
: -----
: Sent from JPTT on my Samsung SM-G935F.
乖乖約分的做法
x^10 - 1 - 10(x-1)
原式 = lim ------------------
x->1 (x-1)^2
(x-1)(x^9+x^8+...+1) - 10(x-1)
= lim ------------------------------
x->1 (x-1)^2
x^9+x^8+...+1-9
= lim --------------- (約掉一個 x-1)
x->1 x-1
(x^9-1)+(x^8-1)+...+(x^2-1)+(x-1)
= lim ---------------------------------
x->1 x-1
(x-1)(x^8+x^7+...+1)+(x-1)(x^7+x^6+...+1)+...+(x-1)(x+1)+(x-1)
= lim --------------------------------------------------------------
x->1 x-1
= lim [(x^8+x^7+...+1)+(x^7+x^6+...+1)+...+(x+1)+1] (約掉第二個 x-1)
x->1
= 9+8+7+...+1 = 45
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標題掛 [中學] 我假定是高三程度
又或者這可以是剛學極限還沒提到羅必達時的做法
全篇只用到 x^n - 1 = (x-1)(x^{n-1}+x^{n-2}+...+x+1) 這條公式
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