Question

lim((a^3+b^3+c^3)/3)^(1/3)的极限x趋向于0

Answer 1

lim((a^x+b^x+c^x)/3)^(1/x)

=lim[1+(a^x+b^x+c^x)/3-1]^(1/x)

=lim[1+(a^x+b^x+c^x)/3-1]^[1/((a^x+b^x+c^x)/3-1)*((a^x+b^x+c^x)/3-1)*(1/x)]

=lime^{[(a^x+b^x+c^x)/3-1]/x}

=lime^[(a^xlna+b^xlnb+c^xlnc)/3]

=[e^(lna+lnb+lnc)]^1/3

=(abc)^1/3

=3次根号(abc).