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We are given a piecewise function \( f(x) \) defined as follows:

\[
f(x) = 
\begin{cases} 
10 - x - x^2 & \text{if } x \leq 4 \\
2x - 5 & \text{if } x > 4 
\end{cases}
\]

We are asked to calculate three limits involving this function:

1. \( \lim_{x \to 4^-} f(x) \) (the left-hand limit as \( x \) approaches 4),
2. \( \lim_{x \to 4^+} f(x) \) (the right-hand limit as \( x \) approaches 4),
3. \( \lim_{x \to 4} f(x) \) (the two-sided limit as \( x \) approaches 4).

### Step 1: Left-hand limit \( \lim_{x \to 4^-} f(x) \)
For the left-hand limit, we are interested in the behavior of \( f(x) \) as \( x \) approaches 4 from values **less than** 4. In this region, the function is defined as \( f(x) = 10 - x - x^2 \).

Substituting \( x = 4 \) into this expression:

\[
f(4^-) = 10 - 4 - 4^2 = 10 - 4 - 16 = -10
\]

Thus, \( \lim_{x \to 4^-} f(x) = -10 \).

### Step 2: Right-hand limit \( \lim_{x \to 4^+} f(x) \)
For the right-hand limit, we consider the behavior of \( f(x) \) as \( x \) approaches 4 from values **greater than** 4. In this region, the function is \( f(x) = 2x - 5 \).

Substituting \( x = 4 \) into this expression:

\[
f(4^+) = 2(4) - 5 = 8 - 5 = 3
\]

Thus, \( \lim_{x \to 4^+} f(x) = 3 \).

### Step 3: Two-sided limit \( \lim_{x \to 4} f(x) \)
For the two-sided limit to exist, the left-hand limit and the right-hand limit must be equal. In this case, we found:

- \( \lim_{x \to 4^-} f(x) = -10 \),
- \( \lim_{x \to 4^+} f(x) = 3 \).

Since these two values are not equal, the two-sided limit **does not exist**.

### Final Answers:
1. \( \lim_{x \to 4^-} f(x) = -10 \)
2. \( \lim_{x \to 4^+} f(x) = 3 \)
3. \( \lim_{x \to 4} f(x) = \text{DNE} \) (does not exist)

Would you like more details on any of these steps or concepts?

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Here are 5 related questions for further exploration:
1. How do you compute the limit of a piecewise function at a different point of discontinuity?
2. What is the significance of one-sided limits in calculus?
3. How can we prove that a function is continuous at a point?
4. How can we interpret the behavior of a function graphically when limits do not exist?
5. What are the criteria for a function to be differentiable at a point?

**Tip:** When limits from the left and right don't match, the function has a **discontinuity** at that point!

Let f(x) = { 10 - x - x^2 if x <= 4, 2x - 5 if x > 4. Calculate the following limits: lim (x -> 4^-) f(x), lim (x -> 4^+) f(x), lim (x -> 4) f(x). Enter 'DNE' if the limit does not exist.

Explore the process of calculating the left-hand and right-hand limits of a piecewise function as x approaches 4. Understand why the two-sided limit does not exist due to discontinuity.

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