To solve for $a$, we need to ensure the continuity of the function $f(x)$ at $x = 3$.

Steps to find $a$:

1. The function is given by:

   5 + \frac{\sin(ax^2 - 9a)}{x - 3} & \text{if } x \neq 3, \\ 8a - 13 & \text{if } x = 3. \end{cases}$$ Since the problem states that $$f(x)$$ is continuous at every point on $$\mathbb{R}$$, particularly at $$x = 3$$, we need to ensure that the limit of $$f(x)$$ as $$x \to 3$$ equals the value of $$f(x)$$ at $$x = 3$$.

2. First, find the limit of $f(x)$ as $x \to 3$ from the expression for $x \neq 3$: $\lim_{x \to 3} \left( 5 + \frac{\sin(ax^2 - 9a)}{x - 3} \right).$ To compute this limit, we need to use L'Hôpital's Rule, as the expression inside involves the indeterminate form $\frac{0}{0}$.

3. The expression $ax^2 - 9a$ simplifies to $a(x^2 - 9) = a(x - 3)(x + 3)$. Therefore, the function becomes: $\frac{\sin(a(x - 3)(x + 3))}{x - 3}.$ As $x \to 3$, the numerator $\sin(a(x - 3)(x + 3)) \to 0$ and the denominator $x - 3 \to 0$, so we can apply L'Hôpital's Rule.

4. Differentiate the numerator and the denominator:

   • The derivative of the denominator $x - 3$ is 1.
   • The derivative of the numerator $\sin(a(x - 3)(x + 3))$ is $\cos(a(x - 3)(x + 3)) \cdot \text{(derivative of the argument)}$, which gives: $\cos(a(x - 3)(x + 3)) \cdot a(2x).$ As $x \to 3$, this simplifies to: $\cos(0) \cdot a(6) = 6a.$

   Therefore, the limit becomes: $\lim_{x \to 3} \left( 5 + \frac{\sin(a(x^2 - 9))}{x - 3} \right) = 5 + 6a.$

5. For continuity, we require that: $\lim_{x \to 3} f(x) = f(3).$ Thus: $5 + 6a = 8a - 13.$

6. Solving for $a$: $5 + 6a = 8a - 13,$ $5 + 13 = 8a - 6a,$ $18 = 2a,$ $a = 9.$

Conclusion:

The value of $a$ is $\boxed{9}$.

Would you like more details on the solution process or have any further questions? Here are some related questions for practice:

1. How do we apply L'Hôpital's Rule in other indeterminate forms?
2. What are the necessary conditions for a function to be continuous?
3. How would you handle a similar problem if the trigonometric function were different (e.g., cosine)?
4. What is the significance of continuity in real-world applications?
5. How can we differentiate trigonometric functions involving composite arguments?

Tip: Always check the limit carefully at the point where the function might be discontinuous. Continuity implies the limit must equal the function's value at that point.