Problem 38 - Essential Calculus

Question

In Exercises \(27-38,\) (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L'Hôpital's Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). \(\lim _{x \rightarrow 1^{+}}\left(\frac{3}{\ln x}-\frac{2}{x-1}\right)\)

Step-by-Step Solution

Answer

The limit of function \(\frac{3}{\ln x}-\frac{2}{x-1}\) as \(x\) approaches 1 from the positive side is -3/2.

Step 1: Direct Substitution to Determine the Indeterminate Form

By direct substitution, attempt to substitute \(x = 1\) into the function: \(\frac{3}{\ln 1}-\frac{2}{1-1}\). Notice that the log of 1 is 0 which leads to a division by 0 in the first term and subtraction of 1 from 1 in the second term also leads to a division by 0. This indicates an indeterminate form of \(\frac{0}{0}\).

Step 2: Applying L'Hopital's Rule

L'Hopital's Rule is applied when the function shows an indeterminate form. Differentiate the numerator and denominator separately: The derivative of \(\frac{3}{\ln x}\) is -\(\frac{3}{x \ln^2 x}\) and the derivative of \(\frac{2}{x-1}\) is \(\frac{2}{{(x-1)}^2}\). After applying L'Hopital's Rule, the function becomes \(\frac{-\frac{3}{x \ln^{2} x}}{\frac{2}{{(x-1)}^{2}}}\).

Step 3: Evaluating the Limit

Evaluate the limit of the derived function as \(x\) approaches 1. Simplify the expression to \(-\frac{3(x-1)^{2}}{2x \ln^{2} x}\). As \(x\) approaches 1 from the positive direction, it results in -3/2.

Step 4: Verifying with Graph

Use a graphing utility to plot the original function \(\frac{3}{\ln x}-\frac{2}{x-1}\) and observe that the function indeed approaches -3/2 as \(x\) approaches 1 from the right.

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool in calculus for resolving limits that yield indeterminate forms, such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). The rule states that if the limit of functions \( f(x) \) and \( g(x) \) as \( x \) approaches a point \( c \) results in an indeterminate form, then the limit of \( \frac{f(x)}{g(x)} \) as \( x \) approaches \( c \) can be calculated by finding the limit of their derivatives instead. In practice, this means:\( \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \), provided the limit on the right exists or is infinite. It's essential, before applying L'Hôpital's Rule, to first determine that the original limit results in an indeterminate form, as was done in Step 1 of the exercise solution.

Limit Evaluation

Evaluating limits is fundamental in understanding the behavior of functions as they approach specific points or infinity. When faced with a challenging limit, it is crucial to explore various methods to find the solution. For instance, if a direct substitution of the point into the function results in a determinate value, then this value is the limit. However, if it leads to an indeterminate form such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), then additional techniques like L'Hôpital's Rule, factoring, rationalization, or limits at infinity need to be employed. The solution to our exercise deployed L’Hôpital’s Rule due to an indeterminate form discovered upon direct substitution, simplifying and differentiating the function to find the limit as \( x \) approaches 1 from the positive side, resulting in the value of -3/2.

Graphing Functions

Graphing functions serves as an insightful visual tool to understand and verify the behavior of functions near certain points. By plotting the function on a coordinate plane, one can observe how the function behaves as it approaches the point of interest from either direction. This graphical representation can confirm whether our analytical solutions align with the visual trend of the function. In the context of our exercise, graphing the given function \( \frac{3}{\ln x}-\frac{2}{x-1} \) using a graphing utility illustrates how the function behaves as \( x \) approaches 1 from the right, substantiating our analytical result of -3/2 as the limit.

Direct Substitution Method

The direct substitution method is a straightforward approach to limit evaluation where the target value is substituted directly into the function. If the substitution yields a finite number, this number is the limit. However, if the substitution results in an undefined expression such as zero divided by zero, or infinity minus infinity, the function presents an indeterminate form and direct substitution is not sufficient to evaluate the limit. In such scenarios, one must resort to alternative methods like L’Hôpital’s Rule, which were used in our exercise when direct substitution failed to provide a conclusive result for the given limit.