※ 引述《ivylan (藍藍)》之銘言： : 4.lim (2x)^x=? : x→0 : 想確認一下答案是不是1 取ln後 xln2x = xln2 + xlnx 這兩項極限都有值 第一項0 第二項 lim xlnx = lim (lnx / 1/x) x->0 = lim (1/x)/(-1/x^2) = 0 取exp就是1 沒錯 x^x^........更複雜的情況我有發過文 : → → : 8.(a)if f(x,y,z) = (x^2+y^2+z^2)^(1/2) , then▽*(▽f) = ? f = r ∂f/∂x = 2x/2r = x/r → → → ▽f = (x/r,y/r,z/r) = (1/r)R r = |R| R : 位置向量 ▽*(▽f) = [(∂/∂x)^2 + (∂/∂y)^2 + (∂/∂z)^2]f (∂/∂x)(∂/∂x)f = (∂/∂x)(x/r) = [r - x(x/r)]/r^2 = (y^2+z^2)/r^3 同理(∂/∂y)^2f = (x^2 + z^2)/r^3 (∂/∂z)^2f = (x^2 + y^2)/r^3 => ▽*(▽f) = 2(x^2 + y^2 + z^2)/r^3 = 2r^2/r^3 = 2/r = 2/√[x^2+y^2+z^2] : → → → : (b)if F(x,y,z) = <x^3+y , y^3+x , z^3+y > , then ▽(▽*F) = ? ▽*F = 3x^2 + 3y^2 + 3z^2 = 3r^2 r^2 = x^2 + y^2 + z^2 ▽(3r^2) = 3 * 2r (x/r , y/r, z/r) = 6 (x, y, z) --

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