Question

How do you find the limit of `(1-4x)^(6/x)` as x approaches 0?

Answer 1

`lim_{x->0}(1 - 4 x)^{6/x} = e^{-24}`

Explanation:

Making `y = -4x` we get

`(1 - 4 x)^{6/x} equiv ((1 + y)^{1/y})^{-4 xx 6}`
then
`lim_{x->0}(1 - 4 x)^{6/x} equiv lim_{y->0} ((1 + y)^{1/y})^{-4 xx 6}`

but

`lim_{y->0} (1 + y)^{1/y} = e`

Finally

`lim_{y->0} ((1 + y)^{1/y})^{-4 xx 6} = e^{-24}`