im asking to find what a is when the function is continuos on a line

To ensure that the function \( g(x) \) is continuous on the real line, we need to make sure that there is no discontinuity at \( x = 0 \), where the two pieces of the piecewise function meet.

The function is defined as:

\[
g(x) =
\begin{cases} 
\frac{3 \sin x}{x}, & \text{if } x < 0 \\
a - 4x, & \text{if } x \geq 0 
\end{cases}
\]

### Steps to Find \( a \) for Continuity:
1. **Continuity at \( x = 0 \)** requires that the limit of \( g(x) \) as \( x \) approaches 0 from the left (i.e., \( x \to 0^- \)) equals the limit as \( x \) approaches 0 from the right (i.e., \( x \to 0^+ \)), and also equals \( g(0) \).

2. **Left-hand limit \( (x \to 0^-) \)**:
   \[
   \lim_{x \to 0^-} g(x) = \lim_{x \to 0^-} \frac{3 \sin x}{x}
   \]
   From the standard limit result:
   \[
   \lim_{x \to 0} \frac{\sin x}{x} = 1
   \]
   Therefore,
   \[
   \lim_{x \to 0^-} g(x) = 3 \cdot 1 = 3
   \]

3. **Right-hand limit \( (x \to 0^+) \)**:
   \[
   \lim_{x \to 0^+} g(x) = \lim_{x \to 0^+} (a - 4x) = a
   \]

4. **Set the left-hand limit equal to the right-hand limit for continuity**:
   \[
   3 = a
   \]

### Conclusion:
For the function \( g(x) \) to be continuous at \( x = 0 \), the constant \( a \) must be \( \mathbf{3} \).

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### 5 Related Questions:
1. What happens if \( a \) is not equal to 3? How does the function behave?
2. Can we also make the function differentiable at \( x = 0 \)?
3. What are the implications of this continuity condition on the overall graph of the function?
4. How does the derivative of the function change at \( x = 0 \)?
5. What other types of piecewise functions require similar continuity conditions?

### Tip:
When determining the continuity of a piecewise function, always equate the limits from both sides at the boundary point where the function pieces meet.

Explore how to determine the constant 'a' to ensure continuity of a piecewise function at the point where its pieces meet. Learn about limits, piecewise functions, and the mathematical steps involved. Suitable for advanced high school students and those interested in calculus concepts.

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