limx属于0（2+x/2-x)的1/x次方的极限是多少.

Question

lim x属于0 （2+x/2-x)的 1/x次方的极限是多少.

Answer 1

lim(x->0) [(2+x)/(2-x)]^(1/x)
=lim [(2-x+2x)/(2-x)]^(1/x)
=lim [1+2x/(2-x)]^(1/x)
=lim [1+1/(2-x)/(2x)]^(1/x)
=lim [1+1/(1/x-1/2)]^(1/x),做这几步是为了要确定是(1+1/x)^x的形式
=lim [1+1/(1/x-1/2)]^{(1/x-1/2)*1/[x(1/x-1/2)]}
=e^lim 1/[x(1/x-1/2)],之前那个应用公式lim(x->0) [(1+1/x)^x]^f(x) = e^[lim f(x)]
=e^lim (1/x)*2x/(2-x)
=e^lim 2/(2-x)
=e^[2/(2-0)]
=e