To evaluate the limit of the given function as $x \to -\infty$, we analyze:

$\lim_{x \to -\infty} \frac{-12x^2 + 8x + 9}{-15x^2 + 2x + 8}$

Step-by-Step Solution:

1. Factor out the highest power of $x$ in both numerator and denominator: The highest power of $x$ is $x^2$. Divide every term in the numerator and denominator by $x^2$:

   $\frac{-12x^2 + 8x + 9}{-15x^2 + 2x + 8} = \frac{-12 + \frac{8}{x} + \frac{9}{x^2}}{-15 + \frac{2}{x} + \frac{8}{x^2}}$

2. Simplify as $x \to -\infty$: As $x \to -\infty$, terms involving $\frac{1}{x}$ and $\frac{1}{x^2}$ approach $0$:

   $\frac{-12 + 0 + 0}{-15 + 0 + 0} = \frac{-12}{-15}.$

3. Simplify the fraction: $\frac{-12}{-15} = \frac{12}{15} = \frac{4}{5}.$

Final Answer:

$\lim_{x \to -\infty} \frac{-12x^2 + 8x + 9}{-15x^2 + 2x + 8} = \frac{4}{5}.$

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Would you like a more detailed explanation or examples of similar problems?

Related Questions:

1. What happens to the limit if the degree of the numerator is higher than the denominator?
2. How would the result change if $x \to \infty$ instead of $x \to -\infty$?
3. Can you show how to evaluate limits when terms include higher-order polynomials?
4. How do limits behave when $x \to 0$ in rational functions?
5. What if the coefficients in the highest degree terms were different signs?

Tip:

Always focus on the highest power of $x$ in rational functions when evaluating limits at infinity!