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We are tasked with finding the limit of the following rational function as \( x \) approaches 4:

\[
\lim_{x \to 4} \frac{6x^2 - 11x + 9}{3x^2 + 3x - 1}
\]

### Step 1: Direct Substitution
Let's first try direct substitution by plugging \( x = 4 \) into both the numerator and denominator.

**Numerator:**
\[
6(4)^2 - 11(4) + 9 = 6(16) - 44 + 9 = 96 - 44 + 9 = 61
\]

**Denominator:**
\[
3(4)^2 + 3(4) - 1 = 3(16) + 12 - 1 = 48 + 12 - 1 = 59
\]

Thus, after direct substitution, we have:
\[
\frac{61}{59}
\]

This is the exact answer, and since it's already in a simplified form, there is no need to perform any additional steps like factoring or using L'Hospital's Rule.

The limit is:
\[
\boxed{\frac{61}{59}}
\]

Would you like more details on this solution, or do you have any questions?

### Related Questions:
1. How do you determine when it's appropriate to directly substitute into a limit?
2. What conditions would require factoring or other methods to find limits instead of direct substitution?
3. Can you always avoid using L'Hospital’s rule for limits involving polynomials?
4. What would happen if direct substitution led to an indeterminate form like \( \frac{0}{0} \)?
5. How does this limit problem differ from one that requires more complex techniques like dividing by the highest power of \( x \)?

### Tip:
Always attempt direct substitution first when solving limits—it's often the quickest and simplest method if no indeterminate form arises.

Find the limit of the rational function lim(x → 4) (6x^2 - 11x + 9) / (3x^2 + 3x - 1).

Learn how to find the limit of the rational function lim(x → 4) (6x^2 - 11x + 9) / (3x^2 + 3x - 1) using direct substitution. This step-by-step guide explains the method and provides the exact solution, suitable for students studying calculus.

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