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The question is asking to find the limit of the given piecewise function \( f(x) \) as \( x \to 2 \). The function is defined as:

\[
f(x) = 
\begin{cases}
4x - 3, & x < 2 \\
(x - 3)^2, & x = 2 \\
3x + 1, & x > 2
\end{cases}
\]

We need to compute \( \lim_{x \to 2} f(x) \), which involves finding the left-hand limit and the right-hand limit at \( x = 2 \) and then checking if they are equal.

### Step 1: Left-hand limit \( \lim_{x \to 2^-} f(x) \)
For \( x < 2 \), the function is given by \( 4x - 3 \). Thus, we compute:
\[
\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (4x - 3) = 4(2) - 3 = 8 - 3 = 5
\]

### Step 2: Right-hand limit \( \lim_{x \to 2^+} f(x) \)
For \( x > 2 \), the function is given by \( 3x + 1 \). Thus, we compute:
\[
\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (3x + 1) = 3(2) + 1 = 6 + 1 = 7
\]

### Step 3: Check if the limits are equal
Since the left-hand limit is 5 and the right-hand limit is 7, the two limits are not equal. Therefore, the two-sided limit does not exist.

Thus, the answer is **(a) Does not exist**.

---

Would you like more details on any part of this solution? Here are some related questions to deepen your understanding:

1. How do we determine the limit for a piecewise function?
2. What does it mean when a two-sided limit does not exist?
3. Can a function still be continuous if the limit does not exist at a particular point?
4. How do left-hand and right-hand limits influence the overall limit of a function?
5. What role does the function definition at \( x = 2 \) play in finding the limit?

**Tip**: Always verify if the left-hand limit equals the right-hand limit to confirm if the overall limit exists.

Let f(x) = 4x - 3, if x < 2; (x - 3)^2, if x = 2; 3x + 1, if x > 2. Then find lim(x -> 2) f(x).

Find the limit of the piecewise function as x approaches 2. This problem focuses on calculating the left-hand and right-hand limits to determine whether the overall limit exists. Suitable for high school calculus students.

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