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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Identify the form of the limit as x approaches infinity. The expression log₂(x) - log₃(x) is of the indeterminate form ∞ - ∞.

To apply l'Hôpital's Rule, we need to rewrite the expression in a form suitable for the rule, such as 0/0 or ∞/∞. Rewrite the expression as a single fraction: (log₂(x) - log₃(x)) = (log(x)/log(2) - log(x)/log(3)).

Combine the terms into a single fraction: (log(x)/log(2) - log(x)/log(3)) = (log(x) * (1/log(2) - 1/log(3))).

Simplify the expression: (log(x) * (1/log(2) - 1/log(3))) = log(x) * ((log(3) - log(2))/(log(2) * log(3))).

Now, evaluate the limit as x approaches infinity: lim_x→∞ (log(x) * ((log(3) - log(2))/(log(2) * log(3)))). Since log(x) approaches infinity as x approaches infinity, the limit depends on the constant factor ((log(3) - log(2))/(log(2) * log(3))).

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Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this context, evaluating the limit as x approaches infinity helps determine the behavior of the function at extreme values.

Logarithmic Functions

Logarithmic functions, such as log₂ x and log₃ x, are the inverses of exponential functions. They are crucial for solving equations involving exponential growth or decay. Understanding the properties of logarithms, including their behavior as x approaches infinity, is key to simplifying and evaluating the limit in the given problem.

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. This rule is particularly useful in the given limit problem to simplify the expression involving logarithms.

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