I was testing if Maple can prove/show limit(a*(1-F(a)), a=infinity,left)=0, where F(.) is a general cumulative distribution function. Though, I doubt Maple has the ability.  However, I come across with some other questions during the exploration:

1. how is maple exactly doing to find limit:
Let's first try a normal distribution case:
 

with(Statistics):
U := RandomVariable(Normal(s,t))  :
limit(CDF(U,x),x=infinity,left)  assuming t>0,t<infinity ;

It returns one even if I drop "t<infinity". But if I put the assuming clause after the definition of U as follows, it fails:

with(Statistics):
U := RandomVariable(Normal(s,t))   assuming t>0,t<infinity :
limit(CDF(U,x),x=infinity,left)  assuming t>0,t<infinity ;

Then, I tried the Chi-square distribution:

U:=RandomVariable(ChiSquare(nu));
limit(CDF(U,x),x=infinity,left) assuming real assuming nu>0,nu<infinity;

It fails. However, CDF(U,infinity) always returns one in the cases above. It seems I have not capture the way Maple derive the limit or how assume facility works.

2. To show this limit proposed in the begining. I come up with an idea to define firstly a general cumulative distribution function F(.) using assume and define command:

assume(F,continuous,F,monotonic);
define(F,F(-infinity)=0,F(infinity)=1);
limit(x*(1-F(x)),x=infinity,left);

But it fails. It think the assume and define statements here are not enough to help Maple to understand F(.) is a CDF. Furtheremore, the information that F(.) is a CDF is not enough for Maple to solve this limit. Right?

Thanks.