Problem 37 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow 0^{+}} \frac{x}{\ln x} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Determine the Form of the Limit

First, substitute \( x = 0^+ \) into the expression \( \frac{x}{\ln x} \). As \( x \to 0^+ \), \( x \to 0 \) and \( \ln x \to -\infty \). Thus, \( \frac{x}{\ln x} \to \frac{0}{-\infty} \), which is an indeterminate form of type \( \frac{0}{-\infty} \).

Step 2: Apply l'Hôpital's Rule

Since the limit is in an indeterminate form, we can apply l'Hôpital's Rule. This requires us to differentiate the numerator and the denominator separately. The derivative of \( x \) is 1, and the derivative of \( \ln x \) is \( \frac{1}{x} \).

Step 3: Compute the New Limit

Apply l'Hôpital's Rule to the limit: \[ \lim _{x \rightarrow 0^{+}} \frac{x}{\ln x} = \lim _{x \rightarrow 0^{+}} \frac{1}{\frac{1}{x}} \]Simplify to:\[ \lim _{x \rightarrow 0^{+}} x \]Evaluate this limit which gives: 0.

Key Concepts

Indeterminate Forms

In calculus, indeterminate forms occur when evaluating limits results in an ambiguous or undefined expression. These forms arise because they don't provide enough information for the limit to be directly evaluated. Common types of indeterminate forms include \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), \( 0 \times \infty \), and others.
L'Hôpital's Rule is often used to tackle these forms, as it can convert these indeterminate expressions into more manageable forms through differentiation. In this way, seemingly complex limits become easier to compute. Detecting the presence of an indeterminate form is the first step before applying this rule, ensuring it is appropriately used.

Limit Calculation

Calculating limits is a fundamental tool in calculus used to analyze the behavior of functions as they approach a certain point. Limits can reveal the value that a function approaches, even if it doesn't explicitly reach that value.
In the exercise, the calculation requires substitution to determine whether the expression forms an indeterminate form. After this initial assessment, limits can be evaluated using various techniques, including algebraic manipulation and rules like l'Hôpital's Rule, which can simplify complex expressions.

Differentiation

Differentiation is the process of finding a derivative, representing the rate of change of a function with respect to its variables. In the context of limit problems, differentiation is key when applying l'Hôpital's Rule. This rule allows us to differentiate the numerator and the denominator independently to resolve indeterminate forms.
For instance, the exercise required differentiating \( x \) to get \( 1 \) and \( \ln x \) to get \( \frac{1}{x} \). These derivatives were then used to simplify the expression under the limit, ultimately making the calculation straightforward.

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is the logarithm to the base \( e \), where \( e \) is an important mathematical constant approximately equal to \( 2.71828 \). It arises in many areas of mathematics due to its natural connection to growth processes and compound interest calculations.
Understanding its behavior is crucial, especially near zero, where \( \ln x \to -\infty \) as \( x \) approaches zero from the positive side. This property was pivotal in identifying the indeterminate form in the exercise, enabling the use of l'Hôpital's Rule to find the solution efficiently.