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In calculus, when evaluating limits, you may encounter expressions that don't provide a clear-cut numerical value at first glance. These are called *indeterminate forms*. Examples of indeterminate forms include:

• \( \frac{0}{0} \)
• \( \frac{\infty}{\infty} \)
• \( \infty - \infty \)
• \( 0 \times \infty \)
• \( 0^0 \)
• \( 1^\infty \)
• \( \infty^0 \)

When you substitute a number into a function and the result is one of these forms, it indicates that you may need additional methods, such as l'Hôpital's Rule, to determine the precise limit. In the given exercise, substituting \( x = 1 \) initially gave us \( \frac{\infty}{0} \) and then \( \frac{0}{0} \) after some algebraic manipulation. These situations signal the presence of an indeterminate form, warranting the use of l'Hôpital's Rule to find the actual limit.

Limits are foundational in calculus, representing the value a function approaches as the input gets arbitrarily close to a particular point. Understanding limits is key to grasping more complicated topics in calculus, such as derivatives and integrals.
Here are a few essential concepts about limits:

• **Limit Notation:** The notation \( \lim_{x \to a} f(x) = L \) means that as \( x \) approaches \( a \), the function \( f(x) \) gets arbitrarily close to \( L \).
• **Existence of Limits:** A limit \( \lim_{x \to a} f(x) \) exists if the values of \( f(x) \) get closer to a single finite number as \( x \) approaches \( a \) from both sides.
• **Infinite Limits:** When a function increases or decreases without bound near a point, the limit can be infinite, represented as \( \pm \infty \).

In the exercise given, we have to evaluate the limit \( \lim_{x \to 1}\left(\frac{1}{x-1}-\frac{x}{\ln x}\right) \). The initial substitution into the limit gives us an indeterminate form, indicating the need to apply additional techniques such as combining fractions or using l'Hôpital's Rule to evaluate it properly.

Differentiation is a fundamental operation in calculus that deals with finding the derivative of a function. A derivative represents the rate at which a function is changing at any given point and is symbolically denoted as \( f'(x) \) or \( \frac{df}{dx} \).
The basic rules of differentiation include:

• **Power Rule:** \( \frac{d}{dx} x^n = nx^{n-1} \)
• **Product Rule:** \( \frac{d}{dx} (uv) = u'v + uv' \)
• **Quotient Rule:** \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2} \)
• **Chain Rule:** \( \frac{d}{dx} f(g(x)) = f'(g(x))g'(x) \)

In the exercise at hand, we use differentiation to handle each step of l'Hôpital's Rule. When faced with an indeterminate form like \( \frac{0}{0} \), l'Hôpital's Rule allows us to take the derivatives of the numerator and the denominator to potentially arrive at a limit we can solve. For example, differentiating the numerator and denominator to resolve \( \lim_{x \to 1} \frac{\ln x - x^2 + x}{(x-1)\ln x} \) eventually leads us to an answer of negative infinity.