Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→∞ log₂ x / log₃ x

Verified step by step guidance

1. Identify the limit to evaluate: lim_{x→∞} (log₂ x) / (log₃ x).
2. Check if the limit is in an indeterminate form. As x approaches infinity, both log₂ x and log₃ x approach infinity, resulting in the form ∞/∞.
3. Since the limit is in the indeterminate form ∞/∞, apply l'Hôpital's Rule, which states that you can take the derivative of the numerator and the derivative of the denominator.
4. Differentiate the numerator: the derivative of log₂ x is (1 / (x ln 2)). Differentiate the denominator: the derivative of log₃ x is (1 / (x ln 3)).
5. Rewrite the limit using the derivatives: lim_{x→∞} (1 / (x ln 2)) / (1 / (x ln 3)) and simplify the expression.