How to find the limit $\lim_{x\to 1^-}\prod_{n=0}^\infty( \frac{1+x^{n+1}}{1+x^n})^{x^n}$?
At first glance, I invoke Bernoulli inequality to estimate the power $x^n$ as $1+n(x-1)\leq x^n\leq 1$. Then we can find some cancellations. However, $1+n(x-1)$ seems to be rough...