f(x) =

x − 6,	     for x ≤ −1
x2 + 1,	     for −1 < x ≤ 1
8 − x,	     for x > 1
Find each limit. (If the limit is infinite, enter '∞' or '-∞', as appropriate. If the limit does not otherwise exist, enter DNE.)
(a)
lim
x→−1−
f(x)

(b)
lim
x→−1+
f(x)

(c)
lim
x→−1
f(x)

(d)
Find
f(−1).

To solve these problems, we will evaluate the piecewise function \( f(x) \) for different limits and for a specific value of \( x \). The function \( f(x) \) is defined as:

\[
f(x) = 
\begin{cases} 
x - 6 & \text{for } x \leq -1 \\
x^2 + 1 & \text{for } -1 < x \leq 1 \\
8 - x & \text{for } x > 1 
\end{cases}
\]

Let's go through each part of the question step-by-step:

### (a) \(\lim_{x \to -1^-} f(x)\)

This represents the limit of \( f(x) \) as \( x \) approaches \(-1\) from the left. For \( x \leq -1 \), the function is defined as \( f(x) = x - 6 \).

So, as \( x \to -1 \) from the left:
\[
f(x) = x - 6 \implies f(-1) = -1 - 6 = -7
\]

Therefore:
\[
\lim_{x \to -1^-} f(x) = -7
\]

### (b) \(\lim_{x \to -1^+} f(x)\)

This represents the limit of \( f(x) \) as \( x \) approaches \(-1\) from the right. For \( -1 < x \leq 1 \), the function is defined as \( f(x) = x^2 + 1 \).

So, as \( x \to -1 \) from the right:
\[
f(x) = x^2 + 1 \implies f(-1) = (-1)^2 + 1 = 1 + 1 = 2
\]

Therefore:
\[
\lim_{x \to -1^+} f(x) = 2
\]

### (c) \(\lim_{x \to -1} f(x)\)

This represents the two-sided limit of \( f(x) \) as \( x \) approaches \(-1\). The limit exists only if the left-hand limit and the right-hand limit are equal.

From parts (a) and (b), we have:
\[
\lim_{x \to -1^-} f(x) = -7 \quad \text{and} \quad \lim_{x \to -1^+} f(x) = 2
\]

Since these two limits are not equal, the limit does not exist.

Therefore:
\[
\lim_{x \to -1} f(x) = \text{DNE}
\]

### (d) \(f(-1)\)

To find the value of \( f(-1) \), we need to check which part of the piecewise function includes \( x = -1 \).

For \( x = -1 \), the condition \( x \leq -1 \) applies, so we use the definition:
\[
f(x) = x - 6
\]

Thus:
\[
f(-1) = -1 - 6 = -7
\]

Therefore:
\[
f(-1) = -7
\]

### Summary of Answers

(a) \(\lim_{x \to -1^-} f(x) = -7\)  
(b) \(\lim_{x \to -1^+} f(x) = 2\)  
(c) \(\lim_{x \to -1} f(x) = \text{DNE}\)  
(d) \(f(-1) = -7\)

Would you like more details on any part, or do you have any questions?

Here are some related questions you might consider:
1. What would \(\lim_{x \to 1^-} f(x)\) be?
2. What would \(\lim_{x \to 1^+} f(x)\) be?
3. How do you find the derivative of this piecewise function?
4. Can you graph this piecewise function?
5. What are the continuity points of \(f(x)\)?

**Tip:** When evaluating limits for piecewise functions, always consider the definition of the function for values approaching from both the left and the right.

Learn how to evaluate limits and function values for a piecewise function including f(x) = x - 6 for x ≤ -1, x^2 + 1 for -1 < x ≤ 1, and 8 - x for x > 1. Detailed solutions for lim x→-1- f(x), lim x→-1+ f(x), lim x→-1 f(x), and f(-1).

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