Compute the limit below.
lim
x
→
0
 
x
2
x
−
3
+
9
x
+
3
 
Give your answer as a fraction, if necessary. For example, if you find that the limit is  
1
2
 , you would enter  
1
2
 .

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.

Compute the limit below. lim(x→0) (x^2) / (x−3+9x+3). Give your answer as a fraction, if necessary.

Discover how to compute the limit of (x^2)/(x−3+9x+3) as x approaches 0. This problem explores algebraic simplifications and limit laws, providing a clear step-by-step solution suitable for high school students.

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