Question

函数f(x)在x=0的邻域内有二阶连续导数,且x→0时limf(x)/x=0,f''(0)=1/4,求x→∞时lim[n

Answer 1

求题目所给极限就是求

lim[f(x)/(x^2)]^(1/2) (数列极限转化为函数极限x=1/n,注意此时x趋向0,我没标出,不好画符号).

现在求根号里面limf(x)/(x^2)的极限 用洛必达即求 limf'(x)/2x

到这一步 在来看已知条件

由x→0时limf(x)/x=0 立即得 f'(0)=0 f(0)=0

又f''(0)=1/4 得lim [f'(x)-f‘(0)]/(x-0)=limf'(x)/x=1/4

于是limf'(x)/2x=1/8 所以lim[f(x)/(x^2)]^(1/2)=根号2/4

x→∞时lim[n^2f(1/n)]^1/2=根号2/4