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Let's examine the piecewise function \( f(x) \) and find the limits requested.

The given function is defined as:

\[
f(x) = \begin{cases} 
-x & \text{if } x \leq -4 \\
16 - x^2 & \text{if } -4 < x < 4 \\
x - 4 & \text{if } x \geq 4
\end{cases}
\]

We need to compute the following limits:

1. \(\lim\limits_{x \to -4^-} f(x)\): The left-hand limit as \(x\) approaches \(-4\).
2. \(\lim\limits_{x \to -4^+} f(x)\): The right-hand limit as \(x\) approaches \(-4\).
3. \(\lim\limits_{x \to -4} f(x)\): The two-sided limit as \(x\) approaches \(-4\).
4. \(\lim\limits_{x \to 4^-} f(x)\): The left-hand limit as \(x\) approaches \(4\).
5. \(\lim\limits_{x \to 4^+} f(x)\): The right-hand limit as \(x\) approaches \(4\).
6. \(\lim\limits_{x \to 4} f(x)\): The two-sided limit as \(x\) approaches \(4\).

### Step 1: Calculate \(\lim\limits_{x \to -4^-} f(x)\)
When \(x \leq -4\), the function is given by \( f(x) = -x \). So as \(x \to -4\) from the left, we have:
\[
\lim\limits_{x \to -4^-} f(x) = -(-4) = 4
\]

### Step 2: Calculate \(\lim\limits_{x \to -4^+} f(x)\)
When \( -4 < x < 4\), the function is \( f(x) = 16 - x^2 \). So as \(x \to -4\) from the right, we have:
\[
\lim\limits_{x \to -4^+} f(x) = 16 - (-4)^2 = 16 - 16 = 0
\]

### Step 3: Calculate \(\lim\limits_{x \to -4} f(x)\)
Since the left-hand and right-hand limits at \(x = -4\) do not agree (\(4 \neq 0\)), the two-sided limit does not exist:
\[
\lim\limits_{x \to -4} f(x) = \text{DNE}
\]

### Step 4: Calculate \(\lim\limits_{x \to 4^-} f(x)\)
When \( -4 < x < 4\), the function is \( f(x) = 16 - x^2 \). So as \(x \to 4\) from the left, we have:
\[
\lim\limits_{x \to 4^-} f(x) = 16 - (4)^2 = 16 - 16 = 0
\]

### Step 5: Calculate \(\lim\limits_{x \to 4^+} f(x)\)
When \(x \geq 4\), the function is \( f(x) = x - 4 \). So as \(x \to 4\) from the right, we have:
\[
\lim\limits_{x \to 4^+} f(x) = 4 - 4 = 0
\]

### Step 6: Calculate \(\lim\limits_{x \to 4} f(x)\)
Since both the left-hand and right-hand limits at \(x = 4\) agree (\(0 = 0\)), the two-sided limit exists:
\[
\lim\limits_{x \to 4} f(x) = 0
\]

### Final Answer:
1. \(\lim\limits_{x \to -4^-} f(x) = 4\)
2. \(\lim\limits_{x \to -4^+} f(x) = 0\)
3. \(\lim\limits_{x \to -4} f(x) = \text{DNE}\)
4. \(\lim\limits_{x \to 4^-} f(x) = 0\)
5. \(\lim\limits_{x \to 4^+} f(x) = 0\)
6. \(\lim\limits_{x \to 4} f(x) = 0\)

Would you like further details or clarification on any part of this solution?

Here are five related questions to expand your understanding:
1. How do you determine if a piecewise function is continuous at a given point?
2. What is the difference between left-hand and right-hand limits?
3. How can you graphically interpret a limit that does not exist?
4. What conditions must be met for a two-sided limit to exist?
5. Can a function have a limit if it is discontinuous at that point?

**Tip:** When evaluating limits of piecewise functions, always check the behavior of the function on both sides of the point in question.

Let f(x) be a piecewise function defined as follows: f(x) = {-x if x <= -4, 16 - x^2 if -4 < x < 4, x - 4 if x >= 4}. Sketch the graph of this function and find the following limits if they exist: lim(x -> -4^-) f(x), lim(x -> -4^+) f(x), lim(x -> -4) f(x), lim(x -> 4^-) f(x), lim(x -> 4^+) f(x), lim(x -> 4) f(x).

This problem involves calculating limits for a piecewise function at x = -4 and x = 4. The function changes behavior across these points, requiring separate limit evaluations from the left and right. Key concepts include the existence of left-hand and right-hand limits, the behavior of quadratic and linear components, and continuity considerations.

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