limx->0 (1-5x)^(1/x)=?

limx->0 (1-5x)^(1/x)=?
limx->0 (1-5x)^(1/x)=?

limx->0 (1-5x)^(1/x)=?
limx->0 (1-5x)^(1/x)
=(1-5x)^[(-1/(5x))*(-5)]
=[(1-5x)^(-1/(5x))]^(-5)
=e^-5
这里用到极限性质limx->0 (1+x)^(1/x)=e

换元:令-5x=y,x=-y/5则原式化为 limx->(1+y)^(-5/y)
当x->0时,y->0.
又因为 limy->0 (1+y)^(-5/y)=limy->0 ((1+y)^(1/y))^-5
由基本极限 limy->0 (1+y)^(1/y)=e
可得:limy->0 ((1+y)^(1/y))^-5=e^-5
即原式=e^-5