Okay, if your figure has n sides then alfa= (360/n)/2= 180/n degrees and so goes to 0 as n goes to infinity. Yes, cot(alfa) = h/(a/2)= 2h/a. And since cot(alpha) goes to infinity as alpha goes to 0, it follows that a goes to 0. You say "h will increase" but that depends on what is being held constant. If "d" is held constant, then, yes, h increases toward d in the limit. I am not clear what you are saying will go o 'H' (I assume you mean 'h').