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!