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The problem asks to determine the coefficient \( a \) such that the function \( f(x) \) is continuous at \( x = 0 \). The function \( f(x) \) is defined as:

\[
f(x) =
\begin{cases}
x^2 + 6a, & x \leq 0 \\
\frac{\sin 6x}{3x}, & x > 0
\end{cases}
\]

To ensure continuity at \( x = 0 \), the following condition must be satisfied:

\[
\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0)
\]

### Step 1: Evaluate \( f(0) \)
From the piecewise definition, for \( x \leq 0 \), we substitute \( x = 0 \) into the expression:

\[
f(0) = 0^2 + 6a = 6a
\]

### Step 2: Evaluate \( \lim_{x \to 0^+} f(x) \)
For \( x > 0 \), the function is \( \frac{\sin 6x}{3x} \). We need to find:

\[
\lim_{x \to 0^+} \frac{\sin 6x}{3x}
\]

Using the standard limit \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \), we apply the same idea here:

\[
\lim_{x \to 0^+} \frac{\sin 6x}{3x} = \frac{6}{3} = 2
\]

### Step 3: Set up the continuity condition
For the function to be continuous at \( x = 0 \), we need:

\[
\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0)
\]

This gives the condition:

\[
6a = 2
\]

### Step 4: Solve for \( a \)

\[
a = \frac{2}{6} = \frac{1}{3}
\]

Thus, the value of \( a \) that makes \( f(x) \) continuous at \( x = 0 \) is \( \frac{1}{3} \).

Would you like more details on any step, or do you have any further questions?

### 5 Related Questions:
1. What is the general definition of continuity at a point?
2. How can the limit of trigonometric functions help solve continuity problems?
3. What are the conditions for a piecewise function to be continuous?
4. How does the Squeeze Theorem apply to limit calculations like \( \lim_{x \to 0} \frac{\sin x}{x} \)?
5. Can a function be continuous but not differentiable? What are examples of such functions?

### Tip:
When checking the continuity of piecewise functions, always ensure the limits from both sides of the point of interest match the function's value at that point.

Tentukan koefisien a agar fungsi f(x) kontinu di x = 0. Fungsi didefinisikan sebagai: f(x) = {x^2 + 6a, x ≤ 0 ; (sin 6x)/(3x), x > 0}

Solve for the coefficient a to make the piecewise function continuous at x = 0. The function involves a quadratic expression and a trigonometric limit. Learn step-by-step with key concepts of limits and continuity.

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