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We have the following problem with the given function \( g(x) \):

\[
g(x) =
\begin{cases} 
x & \text{if } x < 1 \\
3 & \text{if } x = 1 \\
2 - x^2 & \text{if } 1 < x \leq 2 \\
x - 3 & \text{if } x > 2 
\end{cases}
\]

We are asked to find the following limits and values:

### (i) \( \lim_{x \to 1^-} g(x) \)
This is the left-hand limit as \( x \) approaches 1 from the left. For \( x < 1 \), we know that \( g(x) = x \), so:
\[
\lim_{x \to 1^-} g(x) = \lim_{x \to 1^-} x = 1
\]

### (ii) \( \lim_{x \to 1^+} g(x) \)
This is the right-hand limit as \( x \) approaches 1 from the right. For \( 1 < x \leq 2 \), \( g(x) = 2 - x^2 \). Therefore:
\[
\lim_{x \to 1^+} g(x) = 2 - 1^2 = 1
\]

### (iii) \( g(1) \)
From the definition of \( g(x) \), when \( x = 1 \), we have:
\[
g(1) = 3
\]

### (iv) \( \lim_{x \to 2^-} g(x) \)
This is the left-hand limit as \( x \) approaches 2 from the left. For \( 1 < x \leq 2 \), \( g(x) = 2 - x^2 \), so:
\[
\lim_{x \to 2^-} g(x) = 2 - 2^2 = 2 - 4 = -2
\]

### (v) \( \lim_{x \to 2^+} g(x) \)
This is the right-hand limit as \( x \) approaches 2 from the right. For \( x > 2 \), \( g(x) = x - 3 \), so:
\[
\lim_{x \to 2^+} g(x) = 2 - 3 = -1
\]

### (vi) \( \lim_{x \to 2} g(x) \)
Since \( \lim_{x \to 2^-} g(x) = -2 \) and \( \lim_{x \to 2^+} g(x) = -1 \), the two one-sided limits do not match. Therefore, the two-sided limit does not exist:
\[
\lim_{x \to 2} g(x) \text{ does not exist.}
\]

### Summary of Results:
- (i) \( \lim_{x \to 1^-} g(x) = 1 \)
- (ii) \( \lim_{x \to 1^+} g(x) = 1 \)
- (iii) \( g(1) = 3 \)
- (iv) \( \lim_{x \to 2^-} g(x) = -2 \)
- (v) \( \lim_{x \to 2^+} g(x) = -1 \)
- (vi) \( \lim_{x \to 2} g(x) \) does not exist.

Would you like further explanation on any of these limits? Here are some additional questions for you:

1. What conditions must be satisfied for a limit to exist at a point?
2. Why is it important to consider both left-hand and right-hand limits?
3. How does the piecewise nature of \( g(x) \) affect the continuity of the function?
4. Can you find the limit of \( g(x) \) as \( x \to \infty \)?
5. How would you define continuity at a point, based on the function \( g(x) \)?

**Tip:** When evaluating limits, always consider the behavior of the function from both sides of the point to avoid missing any discontinuities.

Explore the limits and continuity of a piecewise function involving left-hand and right-hand limits. This problem involves evaluating limits of a function defined differently across intervals, including one-sided limits and discontinuities. Suitable for calculus students.

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