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The problem asks if the function \( f(x) \) is continuous at \( x = 1 \) and provides a piecewise function:

\[
f(x) = 
\begin{cases} 
x^2 + 1, & \text{if } x < 1 \\
3x, & \text{if } x \geq 1
\end{cases}
\]

To check the continuity of \( f(x) \) at \( x = 1 \), we must verify the following conditions:
1. **\( f(1) \) is defined**.
2. **The left-hand limit exists**: \( \lim_{x \to 1^-} f(x) \).
3. **The right-hand limit exists**: \( \lim_{x \to 1^+} f(x) \).
4. **The function value and the limits must be equal**: \( f(1) = \lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) \).

### Step 1: Find \( f(1) \)
Since \( x = 1 \) falls under the second case (where \( x \geq 1 \)):
\[
f(1) = 3(1) = 3
\]

### Step 2: Compute the left-hand limit (\( x \to 1^- \))
For \( x < 1 \), \( f(x) = x^2 + 1 \), so we find:
\[
\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (x^2 + 1) = 1^2 + 1 = 2
\]

### Step 3: Compute the right-hand limit (\( x \to 1^+ \))
For \( x \geq 1 \), \( f(x) = 3x \), so:
\[
\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (3x) = 3(1) = 3
\]

### Step 4: Check continuity
- \( f(1) = 3 \)
- \( \lim_{x \to 1^-} f(x) = 2 \)
- \( \lim_{x \to 1^+} f(x) = 3 \)

Since \( \lim_{x \to 1^-} f(x) \neq \lim_{x \to 1^+} f(x) \), the function is **not continuous** at \( x = 1 \).

Let me know if you want more details or have any questions!

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Here are 5 related questions to deepen your understanding:
1. What are the conditions for a function to be continuous at a point?
2. How do piecewise functions affect continuity?
3. Can you give an example of a function that is continuous everywhere?
4. What happens when the left-hand and right-hand limits are not equal?
5. How would you define the concept of a limit in simple terms?

**Tip:** When checking continuity at a point in piecewise functions, always carefully analyze each part of the function around that point.

Is the function f(x) = { x^2 + 1, if x < 1; 3x, if x ≥ 1 } continuous at x = 1? Explain.

Check if the piecewise function f(x) = { x^2 + 1 for x < 1; 3x for x ≥ 1 } is continuous at x = 1. Learn about continuity, left-hand and right-hand limits, and how to verify the conditions for a function to be continuous.

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