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To solve this problem, we need to determine the value of \( a \) such that the function \( f(x) \) is continuous at \( x = 0 \).

The given function is:

\[
f(x) = 
\begin{cases} 
\frac{\sqrt{2x + 1} - 1}{\sin(3x)} & \text{for} \ x \neq 0, \\
a & \text{for} \ x = 0.
\end{cases}
\]

### Continuity Condition:
For \( f(x) \) to be continuous at \( x = 0 \), the following condition must hold:
\[
\lim_{x \to 0} f(x) = f(0) = a.
\]
We will first find the limit of \( f(x) \) as \( x \to 0 \) for \( x \neq 0 \), and then set that equal to \( a \).

### Step 1: Find the limit as \( x \to 0 \) for \( x \neq 0 \)
We need to evaluate:
\[
\lim_{x \to 0} \frac{\sqrt{2x + 1} - 1}{\sin(3x)}.
\]

Using the Taylor expansion around \( x = 0 \) for small \( x \):
- \( \sqrt{2x + 1} \approx 1 + x \), and
- \( \sin(3x) \approx 3x \).

Substitute these approximations:
\[
\lim_{x \to 0} \frac{\sqrt{2x + 1} - 1}{\sin(3x)} \approx \lim_{x \to 0} \frac{x}{3x} = \frac{1}{3}.
\]

### Step 2: Set the limit equal to \( a \)
For continuity, we set this limit equal to \( a \):
\[
a = \frac{1}{3}.
\]

Thus, the correct value of \( a \) is \( \frac{1}{3} \), which corresponds to option **c)**.

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Do you want more details or have any questions?

### Related questions:
1. How can you prove a function is continuous using limits?
2. What is the Taylor series for \( \sin(x) \) near zero?
3. Can you explain how to handle indeterminate forms when computing limits?
4. What does it mean for a function to be continuous at a point?
5. How does the small-angle approximation help simplify limit problems?

### Tip:
When faced with a limit involving square roots and trigonometric functions, try using series expansions or L'Hôpital's Rule to simplify the expression.

If f(x) is continuous at x = 0, find the value a such that f(x) is continuous at x = 0, where f(x) = (√(2x + 1) - 1) / sin(3x) for x ≠ 0, and f(x) = a for x = 0.

Determine the value of a to ensure the function f(x) is continuous at x = 0. This problem involves using Taylor expansion and evaluating limits of trigonometric and radical expressions. Suitable for students in Grade 12 or introductory calculus.

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