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?

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.