The expression in the image represents a limit involving an infinite product. Let's break it down step by step.

The general form of the expression is:

$\lim_{x \to 1^-} \prod_{n=0}^{\infty} \left( \frac{1 + x^{n+1}}{1 + x^n} \right)^{x^n}$

Approach:

1. Understand the Limit: The expression contains a product over an infinite number of terms, indexed by $n$. As $x$ approaches 1 from the left (that is, as $x \to 1^{-}$), we will analyze the behavior of each term in the product and the limit as a whole.

2. Simplifying the General Term: Each term in the product has the form:

   $\left( \frac{1 + x^{n+1}}{1 + x^n} \right)^{x^n}$

   For $x$ close to 1, $x^n$ will be very small for large $n$. Therefore, as $x \to 1^{-}$, we expect the terms involving $x^n$ to approach 1, so the product might converge to a specific value.

3. Analysis of the Limit: As $x$ approaches 1 from the left, the terms inside the product become nearly 1 for large $n$, which suggests the product approaches a value related to how the terms behave as $x \to 1^{-}$.

To solve this expression fully, we typically need to apply some advanced series expansions or properties of infinite products.

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