Question

lim x趋于0,（1+x）的1/x次方-e/x的极限

Answer 1

lim(x->0) [(1+x)^(1/x)-e]/x

=lim(x->0) {e^[(1/x)ln(1+x)]-e}/x

=lim(x->0) (1+x)^(1/x)*[1/x(1+x)-ln(1+x)/x^2]

=lim(x->0) (1+x)^(1/x)* lim(x->0) [x-(1+x)ln(1+x)]/(1+x)x^2

=e*lim(x->0) [1-ln(1+x)-1]/[x^2+2x(1+x)]

=-e*lim(x->0) 1/(1+x)(6x+2)

=-e/2