极限高数,lim[sin(x^n)/(sinx)^m]x→0(n,m为正整数)

Question

极限高数,
lim[sin(x^n)/(sinx)^m]
x→0
(n,m为正整数)

Answer 1

∵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m0)[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)]=∞.