设f(t)=lim(x→∞)t(1+2/x)^(x-t),求f'(t)

设f(t)=lim(x→∞)t(1+2/x)^(x-t),求f'(t)
设f(t)=lim(x→∞)t(1+2/x)^(x-t),求f'(t)

设f(t)=lim(x→∞)t(1+2/x)^(x-t),求f'(t)
设f(t)=lim(x→∞)t(1+2/x)^(x-t),求f'(t)
解;∵t(1+2/x)^(x-t)=t[(1+2/x)^(x/2)]²[(1+2/x)^(-t)]
∴f(t)=x→∞limt(1+2/x)^(x-t)=x→∞limt[(1+2/x)^(x/2)]²[(1+2/x)^(-t)]=e²t
故f′(t)=e².