We are tasked with computing the following limit:

$\lim_{x \to 0} \frac{x^2}{x - 3 + 9x + 3}.$

Step-by-step solution:

1. Simplify the denominator: Combine like terms in the denominator:

   $x - 3 + 9x + 3 = x + 9x = 10x.$

   Therefore, the expression becomes:

   $\lim_{x \to 0} \frac{x^2}{10x}.$

2. Simplify the fraction: Cancel out $x$ from the numerator and denominator (note that this is valid as long as $x \neq 0$):

   $\frac{x^2}{10x} = \frac{x}{10}.$

3. Take the limit: Now, compute the limit of $\frac{x}{10}$ as $x \to 0$:

   $\lim_{x \to 0} \frac{x}{10} = \frac{0}{10} = 0.$

Thus, the value of the limit is:

$\boxed{0}.$

Would you like further clarification or details on this solution?

Related questions:

1. What is the limit of $\lim_{x \to 0} \frac{x^3}{x - 5 + 12x + 5}$?
2. How do you find limits that result in indeterminate forms like $\frac{0}{0}$?
3. What happens if the limit as $x \to 0$ for an expression results in $\infty$?
4. Can L'Hopital's rule be applied to this kind of limit problem?
5. What is the limit of $\lim_{x \to 1} \frac{x^2}{2x - 2}$?

Tip: Always check for simplifications in the denominator and numerator before calculating limits.