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To solve this problem, we need to determine the value of \( c \) that ensures the function \( f(x) \) has a limit at \( x = 3 \). The function \( f(x) \) is defined as:

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

To ensure the limit exists at \( x = 3 \), the left-hand limit (as \( x \) approaches 3 from the left) must equal the right-hand limit (as \( x \) approaches 3 from the right), and both must equal \( f(3) \).

### Step 1: Left-hand limit as \( x \to 3^- \)
For \( x < 3 \), we use the expression \( f(x) = ax + 2 \).

\[
\lim_{x \to 3^-} f(x) = a(3) + 2 = 3a + 2
\]

### Step 2: Right-hand limit as \( x \to 3^+ \)
For \( x \geq 3 \), we use the expression \( f(x) = x^2 + c \).

\[
\lim_{x \to 3^+} f(x) = 3^2 + c = 9 + c
\]

### Step 3: Equate the limits
For \( f(x) \) to have a limit at \( x = 3 \), the left-hand and right-hand limits must be equal:

\[
3a + 2 = 9 + c
\]

### Step 4: Solve for \( c \) in terms of \( a \)
Rearrange the equation to solve for \( c \):

\[
c = 3a - 7
\]

Thus, the value of \( c \) in terms of \( a \) that ensures \( f(x) \) has a limit at \( x = 3 \) is \( c = 3a - 7 \).

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Would you like further details on any part of this solution? 

Here are five related questions to explore:

1. What if \( a = 1 \)? What value of \( c \) would make the limit exist at \( x = 3 \)?
2. How would we determine \( a \) and \( c \) if we wanted \( f(x) \) to be continuous at \( x = 3 \)?
3. What happens if we change the value of 2 in \( ax + 2 \) to another constant? How does this affect \( c \)?
4. What is the value of \( f(3) \) in terms of \( c \) and \( a \)?
5. How can we confirm continuity if \( a \) and \( c \) are already determined values?

**Tip:** For a piecewise function to be continuous, ensure that the left and right limits at a boundary point are equal and match the function's value at that point.

Tentukan koefisien c agar f(x) memiliki limit di x = 3, dengan fungsi f(x) = {ax + 2, x < 3; x^2 + c, x ≥ 3}.

Learn how to determine the coefficient c for the piecewise function f(x) = {ax + 2, x < 3; x^2 + c, x ≥ 3} to ensure the existence of a limit at x = 3. This solution explores limits, continuity, and conditions for the left-hand and right-hand limits to be equal.

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