We are given the inequality:

$4x \leq g(x) \leq 2x^4 - 2x^2 + 4$

We need to evaluate:

$\lim_{x \to 1} g(x)$

Step 1: Find the limits of the bounding functions as $x \to 1$

1. The lower bound is $4x$, so let's find its limit as $x \to 1$:

$\lim_{x \to 1} 4x = 4 \cdot 1 = 4$

2. The upper bound is $2x^4 - 2x^2 + 4$, so let's compute its limit as $x \to 1$:

$\lim_{x \to 1} (2x^4 - 2x^2 + 4) = 2(1)^4 - 2(1)^2 + 4 = 2 - 2 + 4 = 4$

Step 2: Apply the Squeeze Theorem

From the inequality $4x \leq g(x) \leq 2x^4 - 2x^2 + 4$, and knowing that both the lower bound and upper bound tend to 4 as $x \to 1$, by the Squeeze Theorem, we conclude:

$\lim_{x \to 1} g(x) = 4$

Conclusion:

$\boxed{4}$

Would you like further details or have any questions? Here are some related questions to explore this topic further:

1. What is the Squeeze Theorem and how does it work?
2. How would the solution change if the bounds of $g(x)$ were different?
3. Can the Squeeze Theorem be applied if the limits of the bounds are not equal?
4. What is the significance of continuity in using the Squeeze Theorem?
5. How do you evaluate limits involving polynomials like $2x^4 - 2x^2 + 4$?

Tip: The Squeeze Theorem is particularly useful when a function is "trapped" between two other functions that approach the same limit.