91Ó°ÊÓ

L'Hôpital's Rule is a powerful tool for evaluating limits that result in indeterminate forms such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). This rule states that if the limits of both the numerator and the denominator of a fraction approach zero or infinity, the limit of the quotient of their derivatives will be equal to the limit of the original fraction. The rule can be expressed as: \[\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}\]provided both \(f'(x)\) and \(g'(x)\) are non-zero near \(c\). This helps to simplify otherwise complicated indeterminate forms. For instance, in the given problem, when evaluating the limit as \(x\) approaches 2 from both sides, we first obtain \(\frac{0}{0}\). By differentiating the numerator and the denominator, we simplify the expression to \(\frac{2x - 3}{3x^2 - 4x}\) and find the values using these derivatives, resulting in a precise solution.

One-sided limits are used to determine the behavior of a function as it approaches a specific point from either the left or the right. These are written as \(\lim_{x \to c^-} f(x)\) for the left-hand limit and \(\lim_{x \to c^+} f(x)\) for the right-hand limit. They are particularly useful when dealing with functions that are not continuous at a point or exhibit different behaviors on either side of a point.

• Left-Hand Limit: What the function is approaching as it comes from values less than the point of interest.
• Right-Hand Limit: What the function is approaching as it comes from values greater than the point of interest.

In the exercise provided, one-sided limits were calculated for \(x \to 2^+\) and \(x \to 2^-\), where both resulted in \(\frac{1}{4}\), indicating that the function is smooth and continuous at \(x = 2\).

Continuous functions are those that do not have any abrupt changes in value, meaning they can be drawn without lifting the pen off the paper. A function \(f(x)\) is continuous at a point \(c\) if the following three conditions are met:

• \(f(c)\) is defined.
• \(\lim_{x \to c} f(x)\) exists.
• \(\lim_{x \to c} f(x) = f(c)\).

In the step-by-step solution, continuity plays a crucial role in determining the limit as \(x\) approaches 2. Because both the left and right-hand limits at \(x = 2\) are equal, the limit exists and the function is continuous at this point. This helps confirm that there are no sudden jumps or breaks at \(x = 2\), leading to the conclusion that the limit is consistently \(\frac{1}{4}\).

Infinity limits describe the behavior of a function as the input approaches very large values, either positively or negatively. These are written as \(\lim_{x \to \infty} f(x)\) or \(\lim_{x \to -\infty} f(x)\). It often signifies a function's asymptotic behavior, which might not land on a specific number.In the exercise, the focus was on one-sided infinite limits as \(x \to 0^+\). When approaching 0 from the positive side, the original denominator approached 0 from the positive side, leading to a vertical asymptote and a resulting limit of \(+\infty\). This is a classic example of an infinite limit where the function heads towards positive infinity rather than a fixed numerical value as it draws near the point from one side. Recognizing these patterns is vital as it signifies the potential of a function growing indefinitely.