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The problem presents two inequalities involving the function \( f(x) \):

1. \( x^4 \leq f(x) \leq x^2 \) for \( x \in [-1, 1] \)
2. \( x^2 \leq f(x) \leq x^4 \) for \( x < -1 \) and \( x > 1 \)

**First Part:** The question asks at what points \( c \) you automatically know \( \lim_{x \to c} f(x) \), and what you can say about the value of the limit at these points.

### Applying the Squeeze Theorem:
The Squeeze Theorem (also called the Sandwich Theorem) is useful here. It states that if:

\[
g(x) \leq f(x) \leq h(x)
\]
and \( \lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L \), then \( \lim_{x \to c} f(x) = L \).

**For \( x \in [-1, 1] \):**
We have the inequality \( x^4 \leq f(x) \leq x^2 \). Now, let’s analyze the limits of these bounding functions as \( x \to c \):
- \( \lim_{x \to c} x^4 = c^4 \)
- \( \lim_{x \to c} x^2 = c^2 \)

Thus, as \( x \to 0 \), both \( x^4 \) and \( x^2 \) approach 0, implying that by the Squeeze Theorem, \( \lim_{x \to 0} f(x) = 0 \).

For any \( c \in (-1, 1) \), the function \( f(x) \) will be squeezed between \( c^4 \) and \( c^2 \), both of which are equal at \( c = \pm 1 \), meaning \( \lim_{x \to c} f(x) = c^4 \).

**For \( x < -1 \) and \( x > 1 \):**
The inequalities are reversed: \( x^2 \leq f(x) \leq x^4 \). Again, let’s analyze the limits for large positive and negative values of \( x \):
- As \( x \to \infty \), both \( x^2 \) and \( x^4 \) approach infinity, implying that \( f(x) \to \infty \) as well.
- Similarly, as \( x \to -\infty \), both \( x^2 \) and \( x^4 \) approach infinity.

Thus, for large positive and negative values of \( x \), the function diverges to infinity.

### Answer:
- For \( c = 0 \), we know \( \lim_{x \to 0} f(x) = 0 \).
- For \( c \in [-1, 1] \), the limit \( \lim_{x \to c} f(x) \) is \( c^4 \).
- For \( |x| > 1 \), as \( x \to \infty \) or \( x \to -\infty \), the limit of \( f(x) \) is infinity.

### Second Part:
The second question asks at which points \( c \), \( \lim_{x \to c} f(x) \) is known. From the first part, we automatically know the limits for:
- \( c = 0 \)
- \( c \in [-1, 1] \) (using the Squeeze Theorem)
- \( |c| > 1 \) (diverges to infinity)

Would you like further details or have additional questions?

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**Relative Questions:**
1. How does the Squeeze Theorem help in determining the limits of a function?
2. What is the significance of the bounding functions \( x^4 \) and \( x^2 \) in this problem?
3. How does the behavior of the function change for \( x \in [-1, 1] \) versus \( x \notin [-1, 1] \)?
4. Why does the limit diverge for large values of \( x \) (both positive and negative)?
5. Can the Squeeze Theorem be applied to cases where the limits of the bounding functions are infinite?

**Tip:** The Squeeze Theorem is particularly useful when dealing with functions bounded by simpler ones whose limits are known at a certain point.

If x^4 ≤ f(x) ≤ x^2 for x in [-1,1] and x^2 ≤ f(x) ≤ x^4 for x < -1 and x > 1, at what points c do you automatically know lim(x→c)f(x)? What can you say about the value of the limit at these points?

This problem explores how the Squeeze Theorem is applied to find the limits of the function f(x) based on the inequalities x^4 ≤ f(x) ≤ x^2 for x in [-1,1], and x^2 ≤ f(x) ≤ x^4 for x < -1 and x > 1. Learn how to determine the limits of f(x) at various points using this theorem.

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