函数连续与导数的一道考研真题第三题我分不太清楚到底为什么选B不选D,

3个回答

  • 选 B).事实上,由于

    lim((x,y)→(0,0))[f(x,y)/(x^2 + y^2)]

    存在,可知应有 f(0,0) = 0.于是

    f'x(0,0) = lim(x→0)[f(x,0) - f(0,0)]/x

    = lim(x→0){[f(x,0) - f(0,0)]/x^2}x = 0,

    同理,f'y(0,0) = 0;进而,由

    lim((Δx,Δy)→(0,0))[f(Δx,Δy) - f(0,0) -f'x(0,0)Δx - f'y(0,0)Δy]/[√(Δx^2 + Δy^2)]

    = lim((Δx,Δy)→(0,0))[f(Δx,Δy)/(Δx^2 + Δy^2)][√(Δx^2 + Δy^2)] = 0,

    得知 f(x,y) 在(0,0) 处可微.