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

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First, identify the form of the limit as x approaches infinity. Both the numerator log₂(x) and the denominator log₃(x) approach infinity, resulting in an indeterminate form ∞/∞.

Since the limit is in the indeterminate form ∞/∞, we can apply l'Hôpital's Rule. This rule states that if the limit of f(x)/g(x) as x approaches a value results in an indeterminate form, then the limit is equal to the limit of their derivatives, f'(x)/g'(x), provided this new limit exists.

Differentiate the numerator: The derivative of log₂(x) with respect to x is 1/(x ln(2)).

Differentiate the denominator: The derivative of log₃(x) with respect to x is 1/(x ln(3)).

Apply l'Hôpital's Rule by taking the limit of the ratio of the derivatives: lim_x→∞ [1/(x ln(2))] / [1/(x ln(3))]. Simplify this expression to find the limit.

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Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this context, we are interested in the behavior of the function as x approaches infinity, which helps us understand the long-term behavior of the logarithmic functions involved.

Logarithmic Functions

Logarithmic functions, such as log₂ x and log₃ x, are the inverses of exponential functions. They are crucial for understanding growth rates and can be transformed using properties of logarithms, such as the change of base formula, which allows us to express logarithms in terms of one another, facilitating limit evaluation.

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 if these forms occur, the limit of the ratio of two functions can be found by taking the derivative of the numerator and the derivative of the denominator, simplifying the evaluation of the limit.

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