第六题怎么做,求极限?
2个回答
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x->0
(1+x)^(1/x)
=e^[ln(1+x)/x]
=e^{ [x -(1/2)x^2 +o(x^2)]/x }
=e^ [1 -(1/2)x +o(x)]
lim(x->0) [(1+x)^(1/x) -e ]/x
=lim(x->0) { e^[1- (1/2)x] -e }/x
=lim(x->0) e. { e^[-(1/2)x] -1 }/x
=lim(x->0) e. [ -(1/2)x ]/x
=-(1/2)e
(1+x)^(1/x)
=e^[ln(1+x)/x]
=e^{ [x -(1/2)x^2 +o(x^2)]/x }
=e^ [1 -(1/2)x +o(x)]
lim(x->0) [(1+x)^(1/x) -e ]/x
=lim(x->0) { e^[1- (1/2)x] -e }/x
=lim(x->0) e. { e^[-(1/2)x] -1 }/x
=lim(x->0) e. [ -(1/2)x ]/x
=-(1/2)e
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