A very difficult task that...

And the question regarding Mathcad here is ... which one?
Whether Mathcad's symbolics is capable of simplifying the term b_n(t) accordingly and specifying the limit value?
I would guess - with a high probability “no”.

By the way, there seems to be something wrong with your statement, as you can easily see if you look at t=0.
For t=0, b_n=2ⁿ (i.e. grows for n-->oo over all limits). However, when inserted into the limit function you specified, t=0 gives the value 1.

The term you name for b_n could be simplified to b_n=sin(t)/sin(t/2ⁿ) by dividing b_n by sin(t/2ⁿ) and continuously applying the theorem of addition -> 2*cos(t/2^k)*sin(t/2^k) = sin(t/2^k-1).

But also this simplified b_n=sin(t)/sin(t/2ⁿ) goes beyond all limits if n->oo, doesn't it?

I can't see a limit of (2*cos(t)+1)/3 here.

Maybe it helps if you post a picture of the original task or provide a link.