Step-by-step Solution:

To determine the value of \( a \) such that \( f(x) \) is continuous at \( x = 3 \), we use the definition of continuity: \[ \lim_{x \to 3^-} f(x) = \lim_{x \to 3^+} f(x) = f(3) \] Step 1: Left-Hand Limit (\( x \to 3^- \)) \[\] For \( x < 3 \), the function is given by: \[ f(x) = x^2 - 1 \] Taking the left-hand limit: \[ \lim_{x \to 3^-} f(x) = 3^2 - 1 = 9 - 1 = 8 \] Step 2: Right-Hand Limit (\( x \to 3^+ \)) \[\] For \( x \geq 3 \), the function is given by: \[ f(x) = 2ax \] Taking the right-hand limit: \[ \lim_{x \to 3^+} f(x) = 2a(3) = 6a \] Step 3: Apply Continuity Condition \[\] Since \( f(x) \) is continuous at \( x = 3 \), we equate the limits: \[ 8 = 6a \] Solving for \( a \): \[ a = \frac{8}{6} = \frac{4}{3} \] Thus, the correct answer is \( \frac{4}{3} \) (Option D).