Question

一题极限高数，求解，

lim[sin(x^n)/(sinx)^m]
x→0
(n,m为正整数)

Answer 1

解：∵原式=lim(x->0){[sin(x^n)/(x^n)]*[(x/sinx)^m]*[x^(n-m)]}
=lim(x->0)[sin(x^n)/(x^n)]*lim(x->0)[(x/sinx)^m]*lim(x->0)[x^(n-m)]
=1*(1^m)*lim(x->0)[x^(n-m)] (应用重要极限lim(z->0)(sinz/z)=1)
=lim(x->0)[x^(n-m)]
∴当m<n时，原式=lim(x->0)[x^(n-m)]=0；
当m=n时，原式=lim(x->0)[x^(n-m)]=1；
当m>n时，原式=lim(x->0)[x^(n-m)]=∞。