Problem 35 - 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). \(\lim _{x \rightarrow 1^{+}}(\ln x)^{x-1}\)

Step-by-Step Solution

Answer

The limit as \(x\) approaches 1 from the right of \((\ln x)^{x-1}\) is 1.

Step 1: Direct Substitution

Let's substitute \(x\) with the given value to evaluate \( (\ln x)^{x-1} \). Substituting \(x = 1\), we get \( (\ln 1)^{1 - 1}\) which results into \(0^0\). The form \(0^0\) is an indeterminate form.

Step 2: Using L'Hôpital's Rule

Due to the indeterminate form, application of L'Hôpital's rule is needed, but our expression isn't in the format of a quotient. To apply L’Hôpital’s rule, we have to rewrite the function as a fraction first. So, we transform \( (\ln x)^{x-1}\) into exponential form, as \(e^{(x - 1)\ln(\ln x)}\). Now, taking natural logarithm on both sides to simplify it into a form suitable for L'Hopital's rule, we have the function as \( \ln(e^{(x - 1)\ln(\ln x)}) = (x - 1) \ln(\ln x)\). Now this can be seen as \(\frac{(x - 1) \ln(\ln x)}{1}\). It's the form \(0/0\) at \(x = 1\), which is suitable for L'Hôpital's Rule.

Step 3: Derivative and limit evaluation

Using L'Hôpital’s rule we differentiate the numerator and the denominator. The derivative of the numerator \((x - 1) \ln(\ln x)\), with respect to \(x\) is \( \ln(\ln x) + (x - 1) * \frac{1}{x * \ln x}\). And, the derivative of 1, the denominator, is just 0. Taking the limit as \(x \rightarrow 1\) of the derivative of the numerator, we get \( \lim _{x \rightarrow 1}(\ln(\ln x) + (x - 1) * \frac{1}{x * \ln x}) = \ln(\ln 1) + (1 - 1) * \frac{1}{1 * \ln 1} = 0 + 0 = 0\).

Step 4: Reverse Transformation

The result obtained is for the transformed function. Therefore, we have to take the exponential to reverse the logarithm applied earlier. Therefore, \( e^0 = 1\).

Key Concepts

Indeterminate Forms

When evaluating limits, we often encounter expressions that do not produce a clear result upon initial observation. These are called indeterminate forms, and they signal that a limit cannot be evaluated through direct substitution alone. In our exercise, the direct substitution of the limit as x approaches 1 from the right side results in the expression \(0^0\), which is one of the classic indeterminate forms. Other common examples include \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), \(0 \times \infty\), \({\infty}^0\), \({0}^\infty\), and \(\infty - \infty\). Each of these forms requires special techniques to resolve the actual limit value, as they do not yield an immediate, obvious result. L'Hôpital's Rule is an essential tool for resolving some of these indeterminate forms, specifically \(\frac{0}{0}\) and \(\frac{\infty}{\infty}\).

Limits

The concept of a limit involves determining the value that a function approaches as the input approaches a certain point. In some cases, directly substituting the point into the function yields the limit. However, when faced with an indeterminate form like \(0^0\), special strategies must be deployed to calculate the limit. The goal is to manipulate or transform the original function into a form that can be evaluated. In the given exercise, L'Hôpital's Rule plays a significant role in this process. By transforming the indeterminate form into \(\frac{0}{0}\), we can differentiate both the numerator and denominator separately and evaluate the limit of the resulting function. This form of analysis is pivotal in calculus, as it helps quantify behavior near points that are not readily accessible through plain arithmetic.

Graphing Utility

A graphing utility is a powerful tool that helps visualize the behavior of functions, confirming the analytical results obtained through techniques like L'Hôpital's Rule. In the problem at hand, graphing can be used post-analysis to verify the limit. By inputting the function \((\ln x)^{x-1}\) into a graphing utility, we can observe the behavior of the curve as \(x\) approaches 1 from the right. The graph assists in corroborating whether the limit converges to the value indicated by the analytical methods. It provides a visual representation making it intuitive to understand, especially when the function's behavior is complex to discern solely by numerical calculations. Hence, when grappling with limits and indeterminate forms, a graphing utility not only strengthens comprehension but also serves as a valuable check against algebraic errors.