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The problem requires us to find the coefficient \( c \) such that the function \( f(x) \) has a limit at \( x = 4 \). The function is defined as:

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

To ensure the function has a limit at \( x = 4 \), we need the left-hand limit (as \( x \to 4^- \)) to be equal to the right-hand limit (as \( x \to 4^+ \)) at \( x = 4 \). This requires that:

\[
\lim_{x \to 4^-} f(x) = \lim_{x \to 4^+} f(x)
\]

### Step 1: Left-hand Limit \( \lim_{x \to 4^-} f(x) \)

For \( x < 4 \), \( f(x) = (3 - x)c \).

\[
\lim_{x \to 4^-} f(x) = \lim_{x \to 4^-} (3 - x)c = (3 - 4)c = -c
\]

### Step 2: Right-hand Limit \( \lim_{x \to 4^+} f(x) \)

For \( x \geq 4 \), \( f(x) = x^2 + 2c \).

\[
\lim_{x \to 4^+} f(x) = \lim_{x \to 4^+} (x^2 + 2c) = 4^2 + 2c = 16 + 2c
\]

### Step 3: Set Left-hand Limit Equal to Right-hand Limit

To have a limit at \( x = 4 \), we set the left-hand and right-hand limits equal to each other:

\[
-c = 16 + 2c
\]

Solving for \( c \):

1. Add \( c \) to both sides:
   \[
   0 = 16 + 3c
   \]

2. Subtract 16 from both sides:
   \[
   -16 = 3c
   \]

3. Divide by 3:
   \[
   c = -\frac{16}{3}
   \]

### Conclusion

The value of \( c \) that makes \( f(x) \) have a limit at \( x = 4 \) is:

\[
c = -\frac{16}{3}
\]

Would you like further details, or have any other questions?

### Related Questions

1. How do you determine the continuity of a piecewise function at a specific point?
2. What is the difference between a limit and continuity?
3. How does one find limits involving piecewise functions in general?
4. Why is it necessary for the left-hand limit to equal the right-hand limit for the existence of a limit?
5. How would the answer change if \( f(x) \) were defined differently for \( x < 4 \) or \( x \geq 4 \)?

### Tip
In piecewise functions, always analyze each section individually when evaluating limits and continuity.

Tentukan koefisien c agar f(x) memiliki limit pada x = 4 dengan f(x) = {(3 - x)c, x < 4; x^2 + 2c, x ≥ 4}

Determine the coefficient c such that f(x) has a limit at x = 4 for the piecewise function f(x) = {(3 - x)c, x < 4; x^2 + 2c, x ≥ 4}. This solution covers key concepts in limits and continuity for piecewise functions. Suitable for students in Grades 10-12.

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