https://file.aiquickdraw.com/mathbot/file/1726622991064zqby4ekl.jpg

The problem in the image asks to find the limit as \( x \to -2 \) of the given rational function:

\[
\lim_{{x \to -2}} \frac{2x^3 - 7x + 6}{5x^2 - 9x - 2}
\]

### Step-by-Step Solution

1. **Direct Substitution**:
   First, substitute \( x = -2 \) into both the numerator and the denominator.

   For the numerator:
   \[
   2(-2)^3 - 7(-2) + 6 = 2(-8) + 14 + 6 = -16 + 14 + 6 = 4
   \]

   For the denominator:
   \[
   5(-2)^2 - 9(-2) - 2 = 5(4) + 18 - 2 = 20 + 18 - 2 = 36
   \]

2. **Evaluating the Limit**:
   After substitution, we get:
   \[
   \lim_{{x \to -2}} \frac{4}{36} = \frac{1}{9}
   \]

Thus, the value of the limit is:
\[
\boxed{\frac{1}{9}}
\]

Would you like further details on any specific part of this solution?

Here are five related questions to practice:
1. What is the limit as \( x \to 1 \) of \( \frac{x^3 - 3x + 2}{2x^2 + 5x - 3} \)?
2. How do you factor cubic polynomials when computing limits?
3. How does the limit change if both numerator and denominator yield zero at \( x = -2 \)?
4. What are indeterminate forms in limits, and how are they resolved?
5. How does L'Hôpital's Rule apply to rational functions?

**Tip:** When direct substitution doesn't work in limits, always check if factoring, simplifying, or using L'Hôpital's Rule might help!

Cari nilai dari limit x → -2 dari (2x^3 - 7x + 6) / (5x^2 - 9x - 2).

This page explains how to find the limit of the rational function (2x^3 - 7x + 6) / (5x^2 - 9x - 2) as x approaches -2 using direct substitution. Learn the key steps in solving this limit problem with detailed explanations.

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