Graph the following function and then find the specified limit. When​ necessary, state that the limit does not exist.
​G(x)equals
left brace Start 2 By 2 Matrix 1st Row 1st Column negative x plus 2 comma 2nd Column for x less than 1 comma 2nd Row 1st Column x squared comma 2nd Column for x greater than 1. EndMatrix 
Find ModifyingBelow lim With x right arrow 1 Upper G left parenthesis x right parenthesis
.

The given piecewise function is defined as:

\[
G(x) =
\begin{cases} 
- x + 2 & \text{for} \, x < 1, \\
x^2 & \text{for} \, x > 1.
\end{cases}
\]

### Step 1: Graphing the Function

We need to graph the two parts of this piecewise function:
1. **For \( x < 1 \):** The function is \( G(x) = -x + 2 \), which is a linear function with slope -1 and intercept at \( (0, 2) \).
2. **For \( x > 1 \):** The function is \( G(x) = x^2 \), which is a parabolic curve opening upwards.

### Step 2: Finding the Limit as \( x \to 1 \)

The goal is to find:
\[
\lim_{x \to 1} G(x).
\]

We need to check both the left-hand and right-hand limits at \( x = 1 \).

- **Left-hand limit** (\( x \to 1^- \)):  
  As \( x \) approaches 1 from the left (\( x < 1 \)), we use the piece \( G(x) = -x + 2 \).  
  \[
  \lim_{x \to 1^-} G(x) = \lim_{x \to 1^-} (-x + 2) = -1 + 2 = 1.
  \]

- **Right-hand limit** (\( x \to 1^+ \)):  
  As \( x \) approaches 1 from the right (\( x > 1 \)), we use the piece \( G(x) = x^2 \).  
  \[
  \lim_{x \to 1^+} G(x) = \lim_{x \to 1^+} (x^2) = 1^2 = 1.
  \]

### Step 3: Conclusion on the Limit

Since both the left-hand and right-hand limits are equal and equal to 1, we can conclude that:
\[
\lim_{x \to 1} G(x) = 1.
\]

Let me know if you would like me to graph the function for you or if you have any questions!

---

Here are 5 related questions to consider:
1. What happens to the limit if the function had a discontinuity at \( x = 1 \)?
2. How would the limit change if the right-hand side of the piecewise function was \( G(x) = x^3 \) for \( x > 1 \)?
3. Can you find the derivative of each piece of the function separately?
4. How would the limit behave if the function was defined differently at \( x = 1 \)?
5. What does it mean when a limit does not exist?

**Tip:** When analyzing piecewise functions, always check both left-hand and right-hand limits to determine whether a limit exists.

Graph the following function and then find the specified limit. When necessary, state that the limit does not exist. G(x) = { -x + 2 for x < 1; x^2 for x > 1 }. Find lim(x -> 1) G(x).

Learn how to graph a piecewise function and determine the limit at a specific point. This problem involves analyzing a linear function for x < 1 and a quadratic function for x > 1, and finding the limit as x approaches 1. Suitable for high school students in Grades 10-12.

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