We are given the inequality:

$8x \leq g(x) \leq 4x^4 - 4x^2 + 8$ and asked to evaluate the limit:

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

To evaluate this limit, we will use the Squeeze Theorem, which is applicable because we have an inequality that bounds $g(x)$ between two functions. The Squeeze Theorem states that if:

$f(x) \leq g(x) \leq h(x)$

and if:

$\lim_{x \to c} f(x) = \lim_{x \to c} h(x) = L$

then:

$\lim_{x \to c} g(x) = L$

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

1. Left bound: $8x$

   $\lim_{x \to 1} 8x = 8(1) = 8$

2. Right bound: $4x^4 - 4x^2 + 8$

   $\lim_{x \to 1} \left( 4x^4 - 4x^2 + 8 \right) = 4(1)^4 - 4(1)^2 + 8 = 4 - 4 + 8 = 8$

Step 2: Apply the Squeeze Theorem.

Since:

$8x \leq g(x) \leq 4x^4 - 4x^2 + 8$

and both the left and right bounds approach 8 as $x \to 1$, by the Squeeze Theorem:

$\lim_{x \to 1} g(x) = 8$

Thus, the limit is:

$\boxed{8}$

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

1. How does the Squeeze Theorem work in general?
2. What happens if the two bounding functions have different limits?
3. Can you apply the Squeeze Theorem when the limits of the bounding functions are not equal?
4. How do you evaluate limits of polynomial functions?
5. How can you prove that a function is continuous using the Squeeze Theorem?

Tip: The Squeeze Theorem is particularly useful when the function is difficult to evaluate directly but is bounded by simpler functions whose limits are easier to compute.