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The problem you provided involves a piecewise function \( g(x) \) defined as follows:

\[
g(x) = 
\begin{cases} 
-x + 1 & \text{for } x < 1 \\
x - 1 & \text{for } 1 < x < 2 \\
5 - x^2 & \text{for } x \geq 2 
\end{cases}
\]

We are asked to graph the function and then solve the following questions:

### (a) \( \lim_{x \to 1} g(x) \)
This limit evaluates the behavior of the function as \( x \) approaches 1 from both sides. We need to check:

- As \( x \to 1^- \), the function \( g(x) = -x + 1 \), so:
  \[
  \lim_{x \to 1^-} g(x) = -(1) + 1 = 0
  \]

- As \( x \to 1^+ \), the function \( g(x) = x - 1 \), so:
  \[
  \lim_{x \to 1^+} g(x) = (1) - 1 = 0
  \]

Since both one-sided limits are equal, the limit exists and is:
\[
\lim_{x \to 1} g(x) = 0
\]

### (b) \( g(1) \)
To find \( g(1) \), we check the function definition. There is no explicit value for \( x = 1 \) in the piecewise function, meaning \( g(1) \) is undefined.

### (c) \( \lim_{x \to 2^-} g(x) \)
For \( x \to 2^- \), we use the second part of the piecewise function \( g(x) = x - 1 \), so:
\[
\lim_{x \to 2^-} g(x) = 2 - 1 = 1
\]

### (d) \( \lim_{x \to 2^+} g(x) \)
For \( x \to 2^+ \), we use the third part of the piecewise function \( g(x) = 5 - x^2 \), so:
\[
\lim_{x \to 2^+} g(x) = 5 - (2)^2 = 5 - 4 = 1
\]

### Conclusion
- \( \lim_{x \to 1} g(x) = 0 \)
- \( g(1) \) is undefined.
- \( \lim_{x \to 2^-} g(x) = 1 \)
- \( \lim_{x \to 2^+} g(x) = 1 \)

Thus, the function is continuous at \( x = 2 \), but there is a discontinuity at \( x = 1 \) because \( g(1) \) is not defined.

Would you like me to plot the graph of the function or explain any part in further detail? Here are some related questions to deepen understanding:

1. How can we determine continuity from a piecewise function?
2. What is the difference between a one-sided limit and a two-sided limit?
3. Can the function be made continuous at \( x = 1 \)?
4. How would the graph behave near \( x = 1 \) and \( x = 2 \)?
5. What happens if the limit from one side does not equal the other?

**Tip:** When working with piecewise functions, always check both one-sided limits to determine if a function is continuous at a given point.

Gambarlah grafik fungsi berikut g(x)= -x+1 ,x < 1 g(x)= x-1, 1 < x < 2 g(x)= 5-x^2 , x >= 2 Kemudian carilah nilai (jika ada) dari (a) lim x mendekati 1 g(x) (b) g(1) (c) lim x mendekati 2- g(x) (d) lim x mendekati 2+ g(x)

Learn how to graph a piecewise function and evaluate limits at specific points. This problem focuses on limits, continuity, and evaluating piecewise-defined functions, suitable for students in Grades 10-12.

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