We are tasked with evaluating the following limit:

$\lim_{{x \to \infty}} \frac{7x^2 - 12x}{7x^4 - 10x^3 - 12}$

Step 1: Analyze the degrees of the polynomials.

• The highest degree in the numerator is $x^2$, with the term $7x^2$.
• The highest degree in the denominator is $x^4$, with the term $7x^4$.

Step 2: Divide by the highest power of $x$ in the denominator.

The highest power of $x$ in the denominator is $x^4$. We divide every term in both the numerator and the denominator by $x^4$:

$\lim_{{x \to \infty}} \frac{\frac{7x^2}{x^4} - \frac{12x}{x^4}}{\frac{7x^4}{x^4} - \frac{10x^3}{x^4} - \frac{12}{x^4}}$

Simplifying the terms:

$= \lim_{{x \to \infty}} \frac{\frac{7}{x^2} - \frac{12}{x^3}}{7 - \frac{10}{x} - \frac{12}{x^4}}$

Step 3: Evaluate the limit as $x \to \infty$.

As $x \to \infty$, terms with powers of $x$ in the denominator go to zero. Thus, we are left with:

$= \frac{0 - 0}{7 - 0 - 0} = \frac{0}{7} = 0$

Final Answer:

$\lim_{{x \to \infty}} \frac{7x^2 - 12x}{7x^4 - 10x^3 - 12} = 0$

Would you like more details or have any further questions?

Here are five related questions to explore:

1. How would the result change if the highest degree in the numerator were equal to the denominator?
2. What happens if we take the limit as $x \to -\infty$?
3. Can we use L'Hôpital's Rule to verify the result?
4. How do limits behave for rational functions when the degree of the denominator is higher than the numerator?
5. What other methods can be used to evaluate limits of rational functions?

Tip: Dividing by the highest power of $x$ is a common technique for simplifying limits involving polynomials at infinity.