Find the constant a such that the function is continuous on the entire real line. \[ g(x)=\left\{\begin{array}{ll} \frac{3 \sin x}{x} & \text { if } x<0 \\ a-6 x & \text { if } x \geq 0 \end{array}\right. \] \[ a= \]

Step 1 ：Given the function \(g(x)=\left\{\begin{array}{ll} \frac{3 \sin x}{x} & \text { if } x<0 \\ a-6 x & \text { if } x \geq 0 \end{array}\right.\)

Step 2 ：For a function to be continuous, the limit from the left and the right at any given point must be equal.

Step 3 ：In this case, we need to find the value of 'a' such that the function is continuous at x=0.

Step 4 ：This means that the limit of the function as x approaches 0 from the left (which is the limit of \(\frac{3\sin(x)}{x}\)) must be equal to the limit of the function as x approaches 0 from the right (which is the limit of \(a-6x\)).

Step 5 ：Calculate the limit from the left: \(\lim_{x \to 0^-} \frac{3\sin(x)}{x} = 3\)

Step 6 ：Calculate the limit from the right: \(\lim_{x \to 0^+} a-6x = a\)

Step 7 ：Set the two limits equal to each other: \(3 = a\)

Step 8 ：\(\boxed{a = 3}\) is the constant that makes the function continuous on the entire real line.