Evaluate $$\lim\limits_{x\to1^-}\prod\limits_{n=0}^\infty\left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}.$$ Please provide the value of a-b, where the original answer is in the form of \ln a-b.

Let's think step by step and output the final answer within \boxed{}.
user