x趋向无穷x^3/[x^k-(x+1)^k]=A为非零常数,求k

Question

x趋向无穷x^3/[x^k-(x+1)^k]=A 为非零常数,求k

Answer 1

k>3
因为：limx^3/[x^k-(x+1)^k]
=lim3x²/[kx^(k-1)-k(x+1)^(k-1)]
=lim6x/[k(k-1)x^(k-2)-k(k-1)(x+1)^(k-2)]
=lim6/[k(k-1)(k-2)[x^(k-3)-(x+1)^(k-3)]=A
所以：[x^(k-3)-(x+1)^(k-3)]≠0
即：k≠3
然而k≥3(显然)
所以：k>3