Problem 31 - 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). \(\lim _{x \rightarrow \infty} x^{1 / x}\)

Step-by-Step Solution

Answer

The indeterminate form is of the type \(1^{\infty}\). The limit of the function \(x^{1/x}\) as \(x\) approaches \(\infty\) is 1, which can be confirmed by a graphical representation of the function.

Step 1: Indeterminate Form Identification

Directly substituting \(x = \infty\) in \(x^{1 / x}\), we get \( \infty^{1/\infty}\), which simplifies to the indeterminate form \(1^{\infty}\). This means the limit could take any value and requires further analysis.

Step 2: Evaluate Limit Using L'Hôpital's Rule

Because we have an indeterminate form, we will use L'Hôpital's rule. But first, the equation must be put into a form that suits L'Hôpital's Rule - ratio of two functions both tending to 0 or \(\infty\). Start by taking the natural logarithm (ln) on both sides of the equation: \( \ln(y) = \ln(x^{1/x}) = \frac{1}{x} \ln(x)\). Now differentiate the numerator and denominator of the expression separately (using L'Hôpital's Rule) to get \(\frac{d}{dx} \ln(y) = \frac{d}{dx} \frac{\ln(x)}{x}\). Simplifying gives \( \frac{y'}{y} = \frac{1 - \ln(x)}{x^2}\) where \(y = x^{1/x}\). So, \(y' = \frac{y(1 - \ln(x))}{x^2}\). As \(x \rightarrow \infty\), \(y' \rightarrow 0\) and thus \(y \rightarrow 1\) since \(y > 0\). So the limit of the function as \(x \rightarrow \infty\), is equal to 1.

Step 3: Graphical Verification

The function should be plotted on a graph for visual verification. A graphing tool will show that the function \(x^{1/x}\) approaches the value 1 as \(x\) tends towards \(\infty\), thereby confirming that the limit is indeed 1.

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool used for evaluating limits, especially when dealing with indeterminate forms like \(0/0\) or \(\infty/\infty\). It provides a method to find limits by differentiating the numerator and the denominator of a given function separately.
Here's the process:

• Identify that the function is in an indeterminate form. If it's not, L'Hôpital's Rule cannot be applied directly.
• Differentiate the numerator and the denominator independently.
• Re-evaluate the limit of the resulting function.

Remember, it's critical that the conditions for using L'Hôpital's Rule are precisely met; otherwise, the solution may not be valid. For instance, with the problem involving \(\lim_{x \rightarrow \infty} x^{1/x}\), we identify the indeterminate form \(1^\infty\). This requires us to manipulate terms to facilitate differentiation—such as using natural logarithms—before applying L'Hôpital's Rule. This approach transforms the problem into a solvable ratio suitable for differentiation, leading us to the conclusion that the limit is 1.

Limits at Infinity

Limits at infinity are a fundamental concept in calculus, helping describe the behavior of functions as the variable approaches infinity or negative infinity. This analysis provides insight into the end behavior of functions.
Key aspects include:

• As \(x\) approaches infinity, we often look at the predominant terms. These terms will dominate the behavior of the function.
• In expressions like \(x^{1/x}\), as \(x\) increases, the term \(1/x\) diminishes toward zero. This impacts the whole expression, simplifying how we understand its limit at infinity.

By understanding such limits, we elucidate the asymptotic nature of functions. In the case of \(x^{1/x}\), evaluating the limit reveals it approaches 1 as \(x \to \infty\). This indicates that for very large \(x\), the function stabilizes around 1, often depicting a horizontal asymptote in the function's graphical representation.

Graphical Verification

Graphical verification involves using a graph to visually confirm the results obtained from analytic calculations of limits and behaviors. It's a crucial step for confirming your findings are correct. Here's what to do:

• Plotting the function helps observe how it behaves as \(x\) reaches large values.
• Visual inspection can verify whether the function indeed approaches the predicted limit.

For our limit problem, graphing \(x^{1/x}\) shows that as \(x\) moves towards infinity, the function's value moves closer and closer to 1, like predicted. This graphical approach not only reinforces your solution but can also aid in understanding limits intuitively. It provides a clear depiction of theoretical outcomes, building confidence in utilizing both algebraic and graphical methods in solving calculus problems.