求 lim(x→∞)[1-(3/x)]^x的极限.
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求 lim(x→∞)[1-(3/x)]^x的极限.
![求 lim(x→∞)[1-(3/x)]^x的极限.](/uploads/image/z/17563149-45-9.jpg?t=%E6%B1%82+lim%28x%E2%86%92%E2%88%9E%29%5B1-%283%2Fx%29%5D%5Ex%E7%9A%84%E6%9E%81%E9%99%90.)
lim(x→∞)[1-(3/x)]^x
=lim(x→∞)e^{ln[1-(3/x)]^x}
=e^lim(x→∞)xln[1-(3/x)]
lim(x→∞)xln[1-(3/x)]
=lim(x→∞){ln[1-(3/x)]}/(1/x)
由洛比达法则
=lim(x→∞)[x/(x-3)]*(-3/-x^2)/[1/(-x^2)]
=lim(x→∞)[x/(x-3)])]*-3
=-3
所以lim(x→∞)[1-(3/x)]^x=e^lim(x→∞)xln[1-(3/x)]=e^(-3)
本来很简单的题.在电脑上写出来看起来很复杂.Orz
用笔照这过程写一遍应该知道了.
=lim(x→∞)e^{ln[1-(3/x)]^x}
=e^lim(x→∞)xln[1-(3/x)]
lim(x→∞)xln[1-(3/x)]
=lim(x→∞){ln[1-(3/x)]}/(1/x)
由洛比达法则
=lim(x→∞)[x/(x-3)]*(-3/-x^2)/[1/(-x^2)]
=lim(x→∞)[x/(x-3)])]*-3
=-3
所以lim(x→∞)[1-(3/x)]^x=e^lim(x→∞)xln[1-(3/x)]=e^(-3)
本来很简单的题.在电脑上写出来看起来很复杂.Orz
用笔照这过程写一遍应该知道了.
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