@Jsevillamol 

Why don't you post an executable worksheet . Use the green arrow to upload it.

Download gi.mw

@Axel Vogt 

For the last limit it's a "semi-bug" because with assuming a>1 it works. And also with assuming a>0,a<=1.

P.S. Many limits are computed by series, not by f/g, L'Hospital or such.

CommonPeriod:=(T1::Not(0),T2::Not(0)) -> `if`(type(T2/T1,rational), abs(numer(T2/T1)*T1), infinity);

PER:=proc(f::algebraic,x::name)
description "The period of a fumction f = g(sin(...),cos(...),...)";
local v:=indets( f, specfunc({sin,cos})), T,t,i;
T:=2*Pi/~ (diff(op~(v),x) minus {0});
if T={} then 
   if depends(f,x) then error "Improper expr" else WARNING("Constant expr"); return 0 fi
fi;
if depends(subs(v=~{seq(t[i],i=1..nops(v))}, f), x) or depends(T,x) 
   then  error "Improper expr" fi ;
t:=op(1,T);
for i in T do 
   t:=CommonPeriod(t,i);
   if t=infinity then return t   fi;
od;
t
end:

PER(a*cos(2*x/3+4) + b*sin(5*x/2)*cos(x), x);
PER(sin(x)+cos(Pi*x), x);
                             12 Pi
                            infinity

@Axel Vogt 
It's not normal that Maple does not know

limit(GAMMA(a,x), x=0,right) assuming a>0;

Note that MultiSeries:-limit  knows it. Same for x=infinity and series. Probably in the near future MultiSeries will be part of "regular" Maple.

@Carl Love 

- The compilation could be done via an auxiliary procedure obtained with ToInert or subs. The problem is whether it is worth the effort.

- But you have recursion in Binomial (using thisproc). Of course it has not a serious impact on performance in our case.

@torabi 

You must give the general form of your function because it would be difficult (time consumming) to consider the period of an arbitrary function.

@Carl Love 

You simply have a typo (x =s).

@Carl Love 

Yes, but the name too long.

@Carl Love 

Thank you. It is very instructive.

@digerdiga 

combine(((1-t)^2)^q) assuming 1-t > 0;
   (1-t)^(2*q)

(even for a complex q)

@Carl Love 

I am not able to provide such an example. A good candidate should be
isolve((y^2-x^3-1)*(x-2*y)*(y-2*x) = 0);
but Maple does not solve y^2-x^3-1 = 0 and however the flag is not set; it should be clear (for Maple and for us too) that not all solutions have been found e.g. x=2,y=3 [actually this is the only one which is missing].
The reason I asked the question is that I don't know why _SolutionsMayBeLost   has disappeared. It would be nice to have it back.

@Carl Love 

I don't think you are right about _SolutionsMayBeLost . solve has no reason to set it because the system is very simple.
And as I have said, the flag was set in earlier versions (and the answer was the same).

@boblas 

I have already shown this:

prod( wdeg([1,1,0,1,1,0,etc], [x,y,z,t,u,v,etc]), plex(x,y,z,t,u,v,etc) );

If you want to test it, please come with a concrete polynomial P (or a list of monomials) and provide also the sorted version of P.

@Carl Love 

If the starting point is changed from x to  `if`(x<2,x/2, ln(x/ln(x)))  then 2 iterations are enough for any x>0 with accuracy 4e-10 for evalhf.  And actually the accuracy is better; for evalf[30] ==> 1e-17.

W(W(`if`(x<2,x/2, ln(x/ln(x))),x),x):

But of course this is not important because an iteration of W is probably not much more expensive than ln(ln(x)).

Edit.  W(W(x/2,x),x)  has accuracy 1e-7 in the interval 0.3 .. 10

@Carl Love 

Very nice, vote up!
So, you took x as starting point for iterations. Of course there are better ones, but it has the advantage that the first iteration does not contain ln.