https://file.aiquickdraw.com/mathbot/file/17304983258548b1fxzxm.jpg

To solve the given limit:

\[
\lim_{{x \to 1}} \left( \frac{x}{x - 1} - \frac{1}{\ln x} \right),
\]

we need to determine if we can apply L'Hospital's Rule, which is typically used for indeterminate forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). Let's analyze each term individually as \(x \to 1\).

### Step 1: Analyzing \(\frac{x}{x - 1}\)
As \(x \to 1\), the denominator \((x - 1)\) approaches zero, and the numerator \(x\) approaches 1. This part alone does not yield a simple indeterminate form; however, \(\frac{x}{x - 1}\) tends to infinity as \(x\) approaches 1.

### Step 2: Analyzing \(\frac{1}{\ln x}\)
As \(x \to 1\), \(\ln x\) approaches zero from below (since \(\ln(1) = 0\)), making \(\frac{1}{\ln x}\) tend to \(-\infty\).

### Step 3: Combining Terms
Now, we have two terms:
\[
\frac{x}{x - 1} \to \infty \quad \text{and} \quad \frac{1}{\ln x} \to -\infty.
\]
This creates an \(\infty - \infty\) indeterminate form, suggesting that L'Hospital's Rule might be useful.

### Step 4: Rewriting and Applying L'Hospital's Rule
Rewrite the expression as a single fraction:
\[
\lim_{{x \to 1}} \left( \frac{x}{x - 1} - \frac{1}{\ln x} \right) = \lim_{{x \to 1}} \frac{x \ln x - (x - 1)}{(x - 1) \ln x}.
\]

Now, we can try to apply L'Hospital's Rule by differentiating the numerator and denominator with respect to \(x\). Differentiating:
1. **Numerator:** The derivative of \(x \ln x - (x - 1)\) is \(\ln x + 1 - 1 = \ln x\).
2. **Denominator:** The derivative of \((x - 1) \ln x\) is \(\ln x + \frac{x - 1}{x} = \ln x + \frac{1 - 1}{1} = \ln

Find the limit. Use L'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If L'Hospital's Rule doesn't apply, explain why.

\[\lim_{{x \to 1}} \left( \frac{x}{x - 1} - \frac{1}{\ln x} \right)\]

Learn how to evaluate the limit \(\lim_{{x \to 1}} \left( \frac{x}{x - 1} - \frac{1}{\ln x} \right)\) by applying L'Hospital's Rule where appropriate. This problem covers limits, indeterminate forms, and natural logarithmic functions, suitable for college-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation