推 kanx:L兄 這一題考不考慮用級數解來作? 220.133.130.61 05/23 17:24
推 lukechen90:還是算的出來阿= =,只是慢了點XD 219.68.232.104 05/23 22:42
推 lukechen90:不過我佩服此PO的勇氣,做好久= = 219.68.232.104 05/23 22:53
※ 引述《kusorz (^~^)》之銘言：
: 1-cosx + tanx-sinx
: lim (______ _______ )
: x→0 x^2 x^3
1 - cosx tanx - sinx
lim (---------- + -------------)
x→0 x^2 x^3
(x)(1 - cosx) + (tanx - sinx)
= lim -------------------------------
x→0 x^3
x - (x)(cosx) + tanx - sinx
= lim -----------------------------
x→0 x^3
1 - cosx + (x)(sinx) + (secx)^2 - cosx
= lim ----------------------------------------
x→0 (3)(x^2)
1 - 2cosx + (x)(sinx) + (secx)^2
= lim ----------------------------------
x→0 (3)(x^2)
2sinx + sinx + (x)(cosx) + (2)(secx)(secx)(tanx)
= lim --------------------------------------------------
x→0 6x
3sinx + (x)(cosx) + (2)((secx)^2)(tanx)
= lim -----------------------------------------
x→0 6x
3sinx (x)(cosx) (2)((secx)^2)(tanx)
= lim ----- + lim --------- + lim ---------------------
x→0 6x x→0 6x x→0 6x
sinx cosx 1 tanx
= lim ------ + lim ------ + lim ((secx)^2)(---)(----)
x→0 2x x→0 6 x→0 3 x
1 sinx 1 1 tanx
= (---)(lim ----) + --- + (---)(lim ((secx)^2))(lim ----)
2 x→0 x 6 3 x→0 x→0 x
1 1 1
= (---)(1) + --- + (---)(1)(1)
2 6 3
1 1 1
= --- + --- + ---
2 6 3
3 + 1 + 2 6
= ----------- = --- = 1
6 6
--
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