Re: 極限
※ 引述《king911015 (早已放棄愛上你)》之銘言：
: 1. lim (1 - 1/4)(1 - 1/9)x…x(1 - 1/X^2) = ?
: x→∞
: 2. Find lim｛(X^3 - 2X^2 +1)^1/3 - X｝
: x→∞
(x^3 - 2x^2 + 1)^(1/3) - x
((x^3-2x^2+1)^(1/3)-x)((x^3-2x^2+1)^(2/3)+((x^3 -2x^2 +1)^(1/3))(x)+ x^2)
= --------------------------------------------------------------------------
(x^3 - 2x^2 + 1)^(2/3) + ((x^3 - 2x^2 + 1)^(1/3))(x) + x^2
x^3 - 2x^2 + 1 - x^3
= ------------------------------------------------------------
(x^3 - 2x^2 + 1)^(2/3) + ((x^3 - 2x^2 + 1)^(1/3))(x) + x^2
-2x^2 + 1
= ----------------------------------------------------------
(x^3 - 2x^2 + 1)^(2/3) + ((x^3 - 2x^2 + 1)^(1/3))(x) + x^2
-2 + (1/(x^2))
= ------------------------------------------------------------
(1-(2/x)+(1/(x^2)))^(2/3) +((1-(2/x)+(1/(x^2)))^(1/3))(x)+1
lim (x^3 - 2x^2 + 1)^(1/3) - x
x→∞
-2 + (1/(x^2))
= lim ------------------------------------------------------------
x→∞ (1-(2/x)+(1/(x^2)))^(2/3) +((1-(2/x)+(1/(x^2)))^(1/3))(x)+1
-2 + 0
= ----------------------------------
(1-0+0)^(2/3) + (1-0+0)^(1/3) + 1
-2 -2
= ----------- = ---
1 + 1 + 1 3
: n 1
: 3. Find the limit lim Σ ------------- = ?
: √(n^2 + k)
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