To solve the given limit problem:

$\lim_{x \to \infty} \frac{x^7 + 2x^5 - 4}{3x^7 + 12}$

Step-by-Step Solution:

1. Identify the highest power of $x$: The numerator ($x^7 + 2x^5 - 4$) and denominator ($3x^7 + 12$) have $x^7$ as the highest power of $x$.

2. Divide every term by $x^7$: This simplifies the fraction: $\frac{x^7 + 2x^5 - 4}{3x^7 + 12} = \frac{\frac{x^7}{x^7} + \frac{2x^5}{x^7} - \frac{4}{x^7}}{\frac{3x^7}{x^7} + \frac{12}{x^7}}$

   Simplify each term: $= \frac{1 + \frac{2}{x^2} - \frac{4}{x^7}}{3 + \frac{12}{x^7}}$

3. Take the limit as $x \to \infty$: As $x \to \infty$, terms with $x$ in the denominator ($\frac{2}{x^2}$, $\frac{4}{x^7}$, $\frac{12}{x^7}$) approach 0: $\lim_{x \to \infty} \frac{1 + \frac{2}{x^2} - \frac{4}{x^7}}{3 + \frac{12}{x^7}} = \frac{1 + 0 - 0}{3 + 0} = \frac{1}{3}.$

Final Answer:

$\lim_{x \to \infty} \frac{x^7 + 2x^5 - 4}{3x^7 + 12} = \frac{1}{3}.$

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Related Questions:

1. What happens if the numerator and denominator have different highest powers of $x$?
2. How do limits work when $x \to -\infty$ instead?
3. What are other methods for evaluating limits of rational functions?
4. How does dividing by the highest power help simplify the problem?
5. Can this concept be extended to non-polynomial functions?

Tip:

When finding the limit of a rational function as $x \to \infty$, focus on the terms with the highest degree in the numerator and denominator. This simplifies the process.