Problem 55 - Calculus - AP Edition

Question

Make a conjecture about the equations of horizontal asymptotes, if any, by graphing the equation with a graphing utility; then check your answer using L'Hôpital's rule. $$y=(\ln x)^{1 / x}$$

Step-by-Step Solution

Answer

The horizontal asymptote is at \(y = 1\).

Step 1: Plot the graph of the function

Using a graphing utility, plot the graph of the function \(y = (\ln x)^{1/x}\). Observe the behavior as \(x\) approaches large values and as \(x\) approaches 0 from the right. You will notice that as \(x\) increases, the function seems to approach a particular constant value, suggesting the presence of a horizontal asymptote.

Step 2: Make a conjecture about the horizontal asymptote

From the graph, observe that as \(x\) becomes very large, \(y\) seems to approach 1. Thus, you can conjecture that there is a horizontal asymptote at \(y = 1\).

Step 3: Apply L'Hôpital's rule to verify the asymptote

To use L'Hôpital's rule, consider the limit \(\lim_{x \to \infty} (\ln x)^{1/x}\). Write this as an exponential function: \(\lim_{x \to \infty} e^{(1/x) \ln(\ln x)}\). Focus on finding \(\lim_{x \to \infty} \frac{\ln(\ln x)}{x}\). This is an indeterminate form \(\frac{\infty}{\infty}\), so apply L'Hôpital's rule: differentiate the numerator and the denominator to get \(\lim_{x \to \infty} \frac{1/(\ln x)}{1} = \lim_{x \to \infty} \frac{1}{x \ln x} = 0\). Thus, the expression simplifies to \(e^0 = 1\).

Step 4: Conclude about the horizontal asymptote

Since the limit as \(x\) approaches infinity is 1, the horizontal asymptote of the function is \(y = 1\). This matches our conjecture from the graph.

Key Concepts

Horizontal Asymptote

A horizontal asymptote is a horizontal line that a graph approaches as the independent variable either increases or decreases without bound. In simpler terms, it's a line that the graph of the function gets closer and closer to, but never quite reaches.

• When determining horizontal asymptotes, look at the behavior of the function as the variable approaches infinity or negative infinity.
• In this case, when graphing the equation \(y = (\ln x)^{1/x}\), you might notice that as \(x\) becomes very large, \(y\) seems to settle around a constant value, suggesting a horizontal asymptote.
• This gives us a clue that there might be a horizontal asymptote at \(y = 1\), which suggests that as \(x\) approaches infinity, \(y\) approaches 1.

Understanding horizontal asymptotes can help predict the end behavior of the function and is crucial in sketching accurate graphs.

Limits

Limits are a fundamental concept in calculus that describe the behavior of a function as its input approaches a particular value. They provide a way to describe how a function behaves as it gets arbitrarily close to a specific point or as it heads towards infinity.

• To determine the horizontal asymptote using limits, you consider the limit of the function as \(x\) approaches infinity.
• For the function \(y = (\ln x)^{1/x}\), determine \(\lim_{x \to \infty} (\ln x)^{1/x}\), which can provide insights into the behavior of the function at extreme values of \(x\).
• By evaluating this limit, especially using L'Hôpital's rule when you encounter indeterminate forms like \(\frac{\infty}{\infty}\), you can confirm the presence of the horizontal asymptote.

Limits are invaluable for understanding and predicting the behavior of functions at points of interest.

Natural Logarithm

The natural logarithm, denoted as \(\ln x\), is the logarithm to the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. Natural logarithms are widely used in mathematics due to their unique properties.

• The function \(y = (\ln x)^{1/x}\) involves the natural logarithm, which makes the behavior of the function interesting as it combines the properties of both logarithm and exponentiation.
• Important properties of the natural logarithm include how it transforms multiplicative processes into additive ones, and how it relates to the growth processes, which are exponential in nature.
• Understanding \(\ln x\) is crucial when working with limits and asymptotic behavior, especially in calculus where exponential and logarithmic functions frequently appear with differentiation and integration.

The natural logarithm is a key function in mathematics, helping to simplify complex multiplicative relationships and appearing in many natural growth processes and mathematical models.