Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→1⁻ (x/(x-1) - x/(ln x)

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First, identify the form of the limit as x approaches 1 from the left. Substitute x = 1 into the expression (x/(x-1) - x/(ln x)) to check if it results in an indeterminate form like 0/0 or ∞/∞.

Notice that as x approaches 1 from the left, both the terms x/(x-1) and x/(ln x) approach forms that could lead to an indeterminate expression. Specifically, x/(x-1) approaches 1/0⁻ (negative infinity) and x/(ln x) approaches 1/0⁻ (negative infinity).

Since the limit results in an indeterminate form of ∞ - ∞, we can combine the fractions into a single fraction to apply l'Hôpital's Rule. Combine the terms over a common denominator: (x(ln x) - x(x-1)) / ((x-1)(ln x)).

Simplify the numerator: x(ln x) - x(x-1) = x ln x - x² + x. The expression becomes (x ln x - x² + x) / ((x-1)(ln x)).

Apply l'Hôpital's Rule, which is applicable for indeterminate forms like 0/0 or ∞/∞. Differentiate the numerator and the denominator separately with respect to x, and then evaluate the limit again as x approaches 1 from the left.

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주요 개념

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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 question, evaluating the limit as x approaches 1 from the left involves analyzing the behavior of the function near that point.

l'Hôpital's Rule

l'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms, such as 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 simplifies the process of finding limits when direct substitution is not possible.

Natural Logarithm (ln)

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.718. It is a crucial function in calculus, particularly in limits and derivatives. In this question, the presence of ln x in the limit expression indicates that understanding its behavior as x approaches 1 is vital for correctly applying l'Hôpital's Rule.

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.lim_x→1⁻ (x/(x-1) - x/(ln x)