- 1

- 0

as x -> 0 we have

(1) t(x)-> oo

(2) x t(x) -> 0

(3) x t(x) exp(t(x)) -> 0

(4) x t(x)^2 exp(t(x)) -> oo

I have "two extremes" as

t(x) = sqrt(-log x) satisfies (1-3) but x t(x)^2 exp(t(x)) -> 0

t(x) = -log( -x log(x))

satisfies (1-2) and (4), but x t(x) exp(t(x)) -> c >0

I have been searching desperately.