Question

设f(x)=lim(n趋于无穷)n次根号下[1+|x|^3n],求f(x)的

求详细根据，，，，，谢谢

Answer 1

lim(n趋于无穷)n次根号下[1+|x|^3n]=lim e^[(1/n)·ln(1+|x|^3n)].
则
|x|<1时，|x|^3n→0.极限=lim e^[(1/n)·(|x|^3n)]
=e^0
=1.
|x|=1时，极限=lim(n趋于无穷)n次根号下[1+1^3n]=2^lim (1/n)=2^0=1
|x|>1时，极限=lim e^[(1/n)·ln(1+|x|^3n)]
=lim e^[(3ln|x|)·|x|^3n/(1+|x|^3n)]
=lim e^[3ln|x|/(1/|x|^3n +1)]
=e^(3ln|x|)
=|x|^3

Answer 2