I think it has something to do with the sequence being strictly monotone decreasing so the limit exists. But actually finding the limit can be challenging. In this case, the denominator is an exponential and it will swamp any polynomial in the numerator so the limit is 0. I know that's not very rigorous, but it's the best I can do.

I think it has something to do with the sequence being strictly monotone decreasing so the limit exists. But actually finding the limit can be challenging. In this case, the denominator is an exponential and it will swamp any polynomial in the numerator so the limit is 0. I know that's not very rigorous, but it's the best I can do.