Problem 51 - Calculus

Question

Compute \(\lim _{x \rightarrow 0} \frac{\ln (1+x)}{x}\)

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Identify the Type of Limit

The given limit \( \lim _{x \rightarrow 0} \frac{\ln (1+x)}{x} \) is of the indeterminate form \( \frac{0}{0} \) as both the numerator and the denominator approach 0 when \( x \to 0 \). This suggests that L'Hopital's Rule can be used.

Step 2: Apply L'Hopital's Rule

According to L'Hopital's Rule, when you have an indeterminate form like \( \frac{0}{0} \), you can differentiate the numerator and the denominator separately. Differentiate the numerator and the denominator:\[ f(x) = \ln(1+x) \rightarrow f'(x) = \frac{1}{1+x} \]\[ g(x) = x \rightarrow g'(x) = 1 \]

Step 3: Perform L'Hopital's Rule and Solve

Apply L'Hopital's Rule:\[ \lim _{x \rightarrow 0} \frac{\ln (1+x)}{x} = \lim _{x \rightarrow 0} \frac{\frac{1}{1+x}}{1} \]Compute the limit:\[ \lim _{x \rightarrow 0} \frac{1}{1+x} = \frac{1}{1+0} = 1 \]

Step 4: Conclude the Calculation

The limit evaluated through L'Hopital's Rule is found to be 1, therefore the value of \( \lim _{x \rightarrow 0} \frac{\ln (1+x)}{x} \) is 1.

Key Concepts

Indeterminate Forms

When dealing with limits involving functions, you might encounter expressions that initially take the form of \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). These are known as indeterminate forms. They occur because both the numerator and the denominator approach zero or infinity as the variable tends toward a particular value, making the limit direct evaluation impossible at first glance. Identifying indeterminate forms is crucial because they tell us when more work is needed to evaluate a limit accurately.

In our given exercise, the limit \( \lim _{x \rightarrow 0} \frac{\ln (1+x)}{x} \) forms a \( \frac{0}{0} \) situation. This happens because both \( \ln(1+x) \) and \( x \) approach 0 as \( x \to 0 \). Recognizing this informs us that we should apply L'Hopital's Rule to find the limit, rather than substituting and simplifying algebraically. The indeterminate form signifies that further analysis, such as differentiation, is needed to solve the problem.

Limit Evaluation

Evaluating limits is a fundamental concept in calculus that explores the behavior of functions as they approach a specific point. When faced with indeterminate forms, L'Hopital's Rule becomes a powerful tool for limit evaluation, primarily when direct substitution is not applicable.

L'Hopital's Rule states that if the limit of a fraction involves an indeterminate form, the limit of the original fraction can be found by differentiating the numerator and denominator separately and then taking the limit again. This rule is especially useful in resolving \( \frac{0}{0} \) and \( \frac{\infty}{\infty} \) forms.

• Step 1: Identify the limit as \( x \to 0 \) or another point of interest.
• Step 2: Determine if it’s an indeterminate form.
• Step 3: Differentiate the numerator and the denominator.
• Step 4: Re-evaluate the limit using the derivatives of the original expressions.

Applying this sequence to our problem allows the use of L'Hopital's Rule, leading to a successful evaluation of the limit as 1.

Differentiation

Differentiation is a key operation in calculus, essential not only for solving problems involving rates of change but also crucial for applying L'Hopital's Rule in limit evaluation. Differentiation involves finding the derivative of a function, which represents the rate of change of the function's value with respect to changes in its input.

For the exercise in question, differentiation is applied to both the numerator \( \ln(1+x) \) and denominator \( x \) of our original limit form \( \frac{\ln(1+x)}{x} \).

• For \( \ln(1+x) \), use the chain rule to find \( f'(x) = \frac{1}{1+x} \).
• The derivative of \( x \) is simply \( g'(x) = 1 \).

By differentiating these components, L'Hopital’s Rule facilitates the limiting process, transforming the original expression to a simpler limit \( \lim _{x \rightarrow 0} \frac{\frac{1}{1+x}}{1} \) that can be evaluated directly, resulting in the final limit of 1. Understanding differentiation ensures the correct application of these principles for accurate problem-solving in calculus.