当x→∞时 （x/x+2）^3x的极限
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解决时间 2021-03-17 16:08
- 提问者网友：無理詩人
- 2021-03-16 17:05
当x→∞时 （x/x+2）^3x的极限
最佳答案
- 五星知识达人网友：千夜
- 2021-03-16 17:21
lim(x→∞)[x/(x+2)]^3x
=lim(x→∞)[(x+2-2)/(x+2)]^3x
=lim(x→∞)[1-2/(x+2)]^3x
=lim(x→∞)[1-2/(x+2)]^{-6[-(x+2)/2]-6}
=lim(x→∞){[1-2/(x+2)]^[-(x+2)/2]}^(-6){[1-2/(x+2)]^(-6)}
=e^(-6)
=lim(x→∞)[(x+2-2)/(x+2)]^3x
=lim(x→∞)[1-2/(x+2)]^3x
=lim(x→∞)[1-2/(x+2)]^{-6[-(x+2)/2]-6}
=lim(x→∞){[1-2/(x+2)]^[-(x+2)/2]}^(-6){[1-2/(x+2)]^(-6)}
=e^(-6)
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