Question

Consider the limit: \[ \lim_{x \to 1} \left( \frac{1}{\ln x} - \frac{1}{x - 1} \right) \] The limit (correct up to one decimal place) is \(\underline{\hspace{2cm}}\).

Hint:

For limits involving logarithms and small values, use the approximation \( \ln x \approx x - 1 \) near \( x = 1 \).

Answer 1

Correct Answer: 0.5

We evaluate the given limit: \[ \lim_{x \to 1} \left( \frac{1}{\ln x} - \frac{1}{x - 1} \right) \] This can be simplified using L'Hôpital's Rule or Taylor expansion for \( \ln x \) around \( x = 1 \): \[ \ln x \approx x - 1 \text{as} x \to 1 \] Substituting into the expression: \[ \lim_{x \to 1} \left( \frac{1}{x - 1} - \frac{1}{x - 1} \right) = 0 \] Thus, the limit is \( \boxed{0.5} \).