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We are given a piecewise function \( f(x) \) and need to evaluate limits as \( x \to -1 \) and \( x \to 1 \). Let's analyze the function and the limits.

### Function Definition:
\[
f(x) =
\begin{cases}
x + 2, & \text{if } x \leq -1 \\
-x^2 + 2x, & \text{if } -1 < x < 1 \\
\frac{1}{x}, & \text{if } x > 1
\end{cases}
\]

### Limits to evaluate:
- \( \lim_{x \to -1^-} f(x) \)
- \( \lim_{x \to -1^+} f(x) \)
- \( \lim_{x \to 1^-} f(x) \)
- \( \lim_{x \to 1^+} f(x) \)

### a. \( \lim_{x \to -1^-} f(x) \):
For \( x \to -1^- \) (from the left side of \(-1\)), the relevant piece of the function is \( f(x) = x + 2 \) (since \( x \leq -1 \)).

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

### b. \( \lim_{x \to -1^+} f(x) \):
For \( x \to -1^+ \) (from the right side of \(-1\)), the relevant piece is \( f(x) = -x^2 + 2x \) (since \( -1 < x < 1 \)).

\[
\lim_{x \to -1^+} f(x) = -(-1)^2 + 2(-1) = -1 - 2 = -3
\]

### c. \( \lim_{x \to 1^-} f(x) \):
For \( x \to 1^- \) (from the left side of \( 1 \)), the relevant piece is \( f(x) = -x^2 + 2x \) (since \( -1 < x < 1 \)).

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

### d. \( \lim_{x \to 1^+} f(x) \):
For \( x \to 1^+ \) (from the right side of \( 1 \)), the relevant piece is \( f(x) = \frac{1}{x} \) (since \( x > 1 \)).

\[
\lim_{x \to 1^+} f(x) = \frac{1}{1} = 1
\]

### Summary of Limits:
- \( \lim_{x \to -1^-} f(x) = 1 \)
- \( \lim_{x \to -1^+} f(x) = -3 \)
- \( \lim_{x \to 1^-} f(x) = 1 \)
- \( \lim_{x \to 1^+} f(x) = 1 \)

Do you need more details or have any questions?

### Related Questions:
1. How do you determine if a function is continuous at a given point?
2. What are the conditions for a limit to exist at a given point?
3. What is the significance of the left-hand and right-hand limits?
4. How would the function's behavior change if the piecewise conditions were modified?
5. Can you generalize this method for finding limits of other piecewise functions?

### Tip:
When evaluating limits of piecewise functions, always check which piece applies in the neighborhood of the point from both sides (left and right).

Evaluate the following limits of the piecewise function f(x) as x approaches -1 and 1 from both sides.

Explore how to evaluate the left-hand and right-hand limits of a piecewise function at points x = -1 and x = 1. This problem demonstrates key calculus concepts such as limits, continuity, and the behavior of functions around specific points.

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