The problem states that $4x \leq g(x) \leq 2x^4 - 2x^2 + 4$ for all $x$, and asks you to evaluate $\lim_{x \to 1} g(x)$.

To solve this, we can apply the Squeeze Theorem (also known as the Sandwich Theorem), which is helpful when you have a function $g(x)$ trapped between two other functions whose limits you can compute.

Step 1: Identify the bounds of $g(x)$

We are given the inequality: $4x \leq g(x) \leq 2x^4 - 2x^2 + 4$

Step 2: Compute the limits of the bounding functions as $x \to 1$

• First, compute $\lim_{x \to 1} 4x$: $\lim_{x \to 1} 4x = 4(1) = 4$

• Next, compute $\lim_{x \to 1} \left( 2x^4 - 2x^2 + 4 \right)$: $\lim_{x \to 1} \left( 2x^4 - 2x^2 + 4 \right) = 2(1^4) - 2(1^2) + 4 = 2 - 2 + 4 = 4$

Step 3: Apply the Squeeze Theorem

Since: $\lim_{x \to 1} 4x = 4 \quad \text{and} \quad \lim_{x \to 1} \left( 2x^4 - 2x^2 + 4 \right) = 4,$ and $g(x)$ is squeezed between these two functions, by the Squeeze Theorem, we conclude: $\lim_{x \to 1} g(x) = 4$

Final Answer:

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

Would you like a deeper explanation of any part of this solution?

Here are 5 related questions:

1. What is the Squeeze Theorem, and when can it be applied?
2. How do you compute limits for polynomial functions?
3. How does the inequality in the problem define the behavior of $g(x)$?
4. What if the limits of the bounding functions were different? How would the result change?
5. Can you apply the Squeeze Theorem to trigonometric functions?

Tip: Always check the limits of both the lower and upper bounding functions carefully before applying the Squeeze Theorem!