Question

设函数f(x)在0<|x|<1有定义，且满足 x趋近于0时 lim(cosx+f(x)/x)的1/x^2等于e^-2，

求x趋近于0是 limf(x)/x^3 给出结题过程 在线等 今晚的数学作业 多谢了

Answer 1

lim(cosx+f(x)/x)^(1/x^2)等于e^-2，
于是limf(x)/x=0(否则，原极限不存在），limx/f(x)=∞
e^-2=lim(cosx+f(x)/x)^(1/x^2)
=lim[(1-1+cosx+f(x)/x)]^ [1/(cosx-1+f(x)/x)] [(cosx-1+f(x)/x)/x^2)]
底数lim[(cosx+f(x)/x)]^[1/(cosx-1+f(x)/x)]=e,
那么指数-2=lim[(cosx-1+f(x)/x)/x^2)]
=lim[x(cosx-1)+f(x)]/x^3
=lim[-(x^3/2)/x^3]+limf(x)/x^3
=-1/2+limf(x)/x^3
limf(x)/x^3=-2+1/2=-3/2