Use l’Hôpital’s rule to find the limits in Exercises 7–52.
39. lim (x → ∞) (ln 2x - ln(x + 1))

Verified step by step guidance

1. First, rewrite the limit expression to understand its form: \(\lim_{x \to \infty} (\ln(2x) - \ln(x + 1))\).
2. Combine the logarithms using the property \(\ln a - \ln b = \ln \left( \frac{a}{b} \right)\), so the limit becomes \(\lim_{x \to \infty} \ln \left( \frac{2x}{x + 1} \right)\).
3. Focus on the argument of the logarithm: \(\frac{2x}{x + 1}\). To find the limit of the logarithm, first find \(\lim_{x \to \infty} \frac{2x}{x + 1}\).
4. Since direct substitution leads to an indeterminate form \(\frac{\infty}{\infty}\), apply l’Hôpital’s Rule by differentiating numerator and denominator separately: differentiate \$2x\( and \)x + 1\( with respect to \)x$.
5. After finding the limit of the fraction using l’Hôpital’s Rule, substitute this limit back into the logarithm to find the overall limit.

Key Concepts

l’Hôpital’s Rule

l’Hôpital’s Rule is a method for evaluating limits that result in indeterminate forms like 0/0 or ∞/∞. It states that the limit of a ratio of functions can be found by taking the limit of the ratio of their derivatives, provided certain conditions are met.

Properties of Logarithms

Understanding logarithmic properties, such as ln(a) - ln(b) = ln(a/b), helps simplify expressions before applying limit techniques. This simplification can make it easier to identify indeterminate forms or apply l’Hôpital’s Rule effectively.

Limits at Infinity

Evaluating limits as x approaches infinity involves analyzing the behavior of functions for very large values of x. Recognizing dominant terms and growth rates is essential to determine whether the limit converges, diverges, or forms an indeterminate expression.