Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$\lim _{x \rightarrow \infty} \frac{\ln x^{10000}}{x}$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Recognize the Form

First, let's express the given limit: \( \lim _{x \rightarrow \infty} \frac{\ln x^{10000}}{x} \). This simplifies to \( \lim _{x \rightarrow \infty} \frac{10000 \ln x}{x} \). Observe as \( x \rightarrow \infty \), both the numerator \( 10000 \ln x \rightarrow \infty \) and the denominator \( x \rightarrow \infty \), which is an \( \frac{\infty}{\infty} \) indeterminate form.

Step 2: Apply l'Hôpital's Rule

Since the form is indeterminate \( \frac{\infty}{\infty} \), apply l'Hôpital's Rule. Differentiate the numerator and denominator separately: \( \frac{d}{dx}(10000 \ln x) = \frac{10000}{x} \) and \( \frac{d}{dx}(x) = 1 \). l'Hôpital's Rule gives us: \( \lim _{x \rightarrow \infty} \frac{10000 \ln x}{x} = \lim _{x \rightarrow \infty} \frac{10000/x}{1} \).

Step 3: Evaluate the Limit

Now evaluate the new limit: \( \lim _{x \rightarrow \infty} \frac{10000}{x} \). As \( x \rightarrow \infty \), the fraction \( \frac{10000}{x} \) approaches zero.

Key Concepts

Indeterminate Forms

In calculus, when dealing with limits, you may encounter expressions that do not lead to an immediate and clear result. These expressions are known as indeterminate forms. The most common types include

• \( \frac{0}{0} \)
• \( \frac{\infty}{\infty} \)
• \( 0 \cdot \infty \)
• \( \infty - \infty \)

Indeterminate forms require special techniques to evaluate, as they suggest ambiguity rather than a specific value. The form \( \frac{\infty}{\infty} \), like the one in our problem, indicates both parts of the fraction approach infinity, thus not providing a priori information about the limit.
lt is important to identify these forms before applying techniques, like l'Hôpital's Rule, to handle them effectively. This rule is often used to resolve such indeterminate forms by involving differentiation to simplify and evaluate the underlying limit.

Limits in Calculus

The concept of limits is foundational in calculus. A limit helps us understand the behavior of a function as the input approaches a particular point. Notably, it describes the value that a function approaches as the input also approaches a given number, and it is a crucial building block for defining both the derivative and the integral.
In our example, we are interested in finding \( \lim _{x \rightarrow \infty} \log(x^{10000}) / x \). As \( x \to \infty \), the function \( \ln(x^{10000}) \) grows without bound due to the natural logarithm's properties. Therefore, understanding limits allows us to address such questions about what happens as \( x \) becomes very large.

• Limits can approach a finite value or infinity.
• Using limit properties and rules can simplify complex problems.

Grasping limits is essential for subsequent topics in calculus, where they protect the rigor and methodological understanding needed in calculus practices.

Differentiation Techniques

Differentiation techniques are key strategies in calculus used to find the derivative of a function. The derivative represents the function's rate of change and is essentially the slope of the function at any point.
In the exercise, l'Hôpital's Rule requires

• Differentiating the numerator: For \( 10000 \ln x \), the derivative is \( 10000/x \).
• Differentiating the denominator: For \( x \), the derivative is \( 1 \).

This transformation simplifies into a new, more straightforward limit form. Differentiation, thus, helps in converting complex forms into easily evaluable expressions, streamlining the limit evaluation process.
It is essential for students to get comfortable with basic differentiation techniques, such as the power rule, product rule, quotient rule, and chain rule. These tools arm you with the skills needed to apply l'Hôpital's Rule effectively, making the evaluation of indeterminate forms much more manageable.