To evaluate the limit:

$\lim_{x \to \infty} \frac{5x^3 + 6x^2}{6x^3 - 7}$

we can simplify by dividing both the numerator and denominator by $x^3$, the highest power of $x$ present in the expression:

$\lim_{x \to \infty} \frac{5 + \frac{6}{x}}{6 - \frac{7}{x^3}}$

As $x \to \infty$, terms with $\frac{6}{x}$ and $\frac{7}{x^3}$ approach zero, leaving:

$\frac{5}{6}$

Thus, the evaluated limit is:

$\lim_{x \to \infty} \frac{5x^3 + 6x^2}{6x^3 - 7} = \frac{5}{6}$

Would you like more details or have any questions?

Related Questions:

1. What would change if the limit was $x \to -\infty$?
2. How do we handle limits at infinity for expressions with different leading terms?
3. What if the powers of $x$ in the numerator and denominator were different?
4. How does the limit change if we add a constant to the numerator or denominator?
5. How does this concept apply to finding horizontal asymptotes?

Tip: When evaluating limits at infinity, divide by the highest power of $x$ in the denominator to simplify the expression.