Use l’Hôpital’s rule to find the limits in Exercises 7–52.


12. lim (x → ∞) (x - 8x²) / (12x² + 5x)

Verified step by step guidance

1. Identify the limit expression: \(\lim_{x \to \infty} \frac{x - 8x^{2}}{12x^{2} + 5x}\).
2. Check the form of the limit by analyzing the degrees of the numerator and denominator as \(x\) approaches infinity. Both numerator and denominator tend to infinity, so the limit is of the form \(\frac{\infty}{\infty}\), which is an indeterminate form suitable for l'Hôpital's Rule.
3. Apply l'Hôpital's Rule by differentiating the numerator and denominator separately with respect to \(x\): differentiate numerator \(\frac{d}{dx}(x - 8x^{2})\) and denominator \(\frac{d}{dx}(12x^{2} + 5x)\).
4. Write the new limit expression using the derivatives: \(\lim_{x \to \infty} \frac{\frac{d}{dx}(x - 8x^{2})}{\frac{d}{dx}(12x^{2} + 5x)}\).
5. Evaluate the new limit by simplifying the derivatives and then analyzing the behavior as \(x\) approaches infinity to find the limit.

Key Concepts

Limits at Infinity

Limits at infinity describe the behavior of a function as the input grows without bound. Understanding how to evaluate these limits helps determine the end behavior of rational functions, often by comparing the degrees of polynomials in the numerator and denominator.

Indeterminate Forms

Indeterminate forms like ∞/∞ or 0/0 occur when direct substitution in a limit does not yield a clear answer. Recognizing these forms is essential because they signal the need for techniques like l’Hôpital’s rule to evaluate the limit properly.

l’Hôpital’s Rule

l’Hôpital’s rule provides a method to evaluate limits that result in indeterminate forms by differentiating the numerator and denominator separately. Applying this rule simplifies the limit calculation, especially for rational functions where direct substitution is inconclusive.