The expression in the image involves a limit and an infinite product. Let's break it down.

The function $f(x)$ is defined as:

$f(x) = \lim_{n \to \infty} \prod_{n=0}^{n} \left( \frac{x^{2^n} + 1}{x^{2^{n+1}} + 1} \right)^2$

The domain of $x$ is specified as $x \in [0, \infty) \setminus \{1\}$, meaning $x$ can take values in the non-negative real numbers except for $1$. Additionally, it is given that $f(1) = 1$.

Key Steps to Solve:

1. Understanding the Infinite Product: The product given involves terms that compare powers of $x$ in the numerator and denominator, squared. You would have to evaluate how this infinite product behaves as $n \to \infty$.

2. Limit Behavior: The limit as $n \to \infty$ could potentially lead to simplifications, especially because of the powers of $x$. Typically, you would study this behavior for specific values of $x$, such as $x = 0$, $x = 1$, and as $x \to \infty$.

3. Behavior near Singularities: You are instructed that $x = 1$ is a special case, and $f(1) = 1$. Thus, it's important to examine how the function behaves around this point.

Would you like me to work through the detailed analysis for this function, including determining how to approach the limit?

Here are some questions related to this topic:

1. How does the infinite product behave when $x \to 0$?
2. What happens to the infinite product as $x \to \infty$?
3. How would you derive the value of the function at points other than 1?
4. What mathematical tools can help simplify infinite products?
5. How does the exclusion of $x = 1$ affect the analysis of this function?

Tip:

When dealing with infinite products, try evaluating the behavior of the terms as $n$ increases. Often, some form of convergence or divergence helps identify the function's limits.