Question

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow \infty} x^{1 / x} $$

Short answer

The limit is 1.

Step-by-step solution

Identify the Type of Limit

First, recognize that we are looking at the limit \( \lim_{x \to \infty} x^{1/x} \). We begin by considering the base \(x\) raised to the power \(1/x\). We need to identify if this forms an indeterminate expression.

Transform the Expression

To better analyze the limit, transform the expression using a logarithm. We consider \( y = x^{1/x} \) and take the natural logarithm of both sides to get \( \ln(y) = \frac{1}{x} \ln(x) \). This transformation helps in dealing with the indeterminate form by converting the power into a product.

Check for Indeterminate Form

Now, consider the limit of \( \ln(y) \) as \( x \to \infty \): \( \lim_{x \to \infty} \frac{\ln(x)}{x} \). As this is a form of \( \frac{\infty}{\infty} \), we can apply l'Hôpital's Rule.

Apply l'Hôpital's Rule

Apply l'Hôpital's Rule to the limit \( \lim_{x \to \infty} \frac{\ln(x)}{x} \). Differentiate the numerator and the denominator: The derivative of \( \ln(x) \) is \( 1/x \) and the derivative of \( x \) is 1. Thus, we have \( \lim_{x \to \infty} \frac{1/x}{1} = \lim_{x \to \infty} \frac{1}{x} = 0 \).

Evaluate the Original Limit

As \( \ln(y) \to 0 \), it means that \( y \to e^0 = 1 \). Therefore, the original limit \( \lim_{x \to \infty} x^{1/x} = 1 \).

Key concepts

Indeterminate Forms

In calculus, when evaluating limits, you may encounter expressions that seem to take an indeterminate form. Indeterminate forms are expressions where a direct substitution might suggest an answer of infinity, zero, or something else equally confusing that doesn't directly resolve to a specific number. Common indeterminate forms include \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), \( 0\cdot\infty \), and others. Recognizing these forms is crucial because they often enable the use of specific mathematical tools to find the solution.

L'Hôpital's Rule is frequently used to tackle indeterminate forms like \( \frac{\infty}{\infty} \) or \( \frac{0}{0} \). When faced with such forms, it's possible to differentiate the numerator and the denominator separately to evaluate the limit. By correctly identifying indeterminate forms, you can simplify even complex expressions into something more manageable.

Limits in Calculus

Limits are foundational to calculus. They describe the behavior of a function as the input (or \( x \)) approaches a certain value. Limits help in defining critical calculus concepts like derivatives and integrals. Without understanding limits, you can't fully grasp how calculus functions operate and change.

Consider the exercise at hand. Here, we have the limit \( \lim_{x \to \infty} x^{1/x} \). This phrase is asking us to determine what value \( x^{1/x} \) approaches as \( x \) becomes infinitely large. To evaluate this limit effectively, we first needed to transform it using logarithms to demonstrate it can be simplified through other calculus techniques, particularly using L'Hôpital's Rule for indeterminate forms.

Calculating limits is a step-by-step process that usually involves testing substitutions and transformations to reveal the solution. Patience and practice are key in getting comfortable with limits.

Logarithmic Transformations

In many mathematical problems, especially those involving limits and indeterminate forms, logarithmic transformations prove to be incredibly useful. Logarithms allow you to transform products, powers, and roots into sums and simpler components that can be managed more easily. This transformation can significantly ease complicated expressions and allow further manipulations like differentiation.

In the exercise of finding the limit \( \lim_{x \to \infty} x^{1/x} \), a logarithmic transformation is applied by setting \( y = x^{1/x} \) and taking the natural log of both sides, giving \( \ln(y) = \frac{1}{x} \ln(x) \). By doing this, we transformed a tricky power into a straightforward product. This set the stage for applying L'Hôpital's Rule on the form \( \frac{\ln(x)}{x} \), ultimately simplifying our calculations.

Mastering logarithmic transformations can make solving complex calculus problems much more manageable, highlighting its importance as an essential tool in the student’s mathematical toolkit.