The problem is:

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

Step-by-step solution:

1. Identify the indeterminate form:

• As $x \to 1$, the first term $\frac{x}{x - 1}$ approaches $\frac{1}{0}$ (infinity).
• The second term $\frac{1}{\ln x}$ also approaches infinity since $\ln x \to 0$.

Thus, the overall expression is in the indeterminate form $\infty - \infty$. We must simplify further.

2. Combine the terms into a single fraction:

Write the expression with a common denominator: $\frac{x}{x - 1} - \frac{1}{\ln x} = \frac{x \ln x - (x - 1)}{(x - 1) \ln x}.$

Now the limit becomes: $\lim_{x \to 1} \frac{x \ln x - (x - 1)}{(x - 1) \ln x}.$

3. Expand and analyze the numerator:

• As $x \to 1$, expand $\ln x$ using a Taylor series: $\ln x \approx (x - 1) - \frac{(x - 1)^2}{2} + \mathcal{O}((x - 1)^3).$

Thus: $x \ln x = x \big((x - 1) - \frac{(x - 1)^2}{2} + \cdots \big).$

For small $x - 1$, $x \approx 1 + (x - 1)$, so: $x \ln x \approx (x - 1) - \frac{(x - 1)^2}{2} + \cdots.$

The numerator becomes: $x \ln x - (x - 1) \approx -\frac{(x - 1)^2}{2} + \cdots.$

4. Simplify the denominator:

The denominator is: $(x - 1) \ln x \approx (x - 1)((x - 1) - \frac{(x - 1)^2}{2} + \cdots).$

This simplifies to: $(x - 1)^2 - \frac{(x - 1)^3}{2} + \cdots.$

5. Simplify the fraction:

Divide the numerator by the denominator: $\frac{x \ln x - (x - 1)}{(x - 1) \ln x} \approx \frac{-\frac{(x - 1)^2}{2}}{(x - 1)^2}.$

Cancel $(x - 1)^2$: $\approx -\frac{1}{2}.$

Final Answer:

$\lim_{x \to 1} \left( \frac{x}{x - 1} - \frac{1}{\ln x} \right) = -\frac{1}{2}.$

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Related Questions:

1. What is the general approach to solving $\infty - \infty$ limits?
2. How do Taylor series expansions simplify indeterminate forms?
3. Can L'Hôpital's Rule be applied to this problem? If yes, how?
4. How does $\ln x$ behave as $x \to 1$?
5. What are some other examples of combining fractions to resolve limits?

Tip:

When faced with $\infty - \infty$ forms, always look for ways to combine terms into a single fraction for simplification.