Question

Choose the most appropriate option. \(\lim_{x \to 1} x^{(1-x)}\) is equal to:

• A. 0
• B. 3
• C. 1/e
• D. $\infty$

Hint:

For indeterminate forms like \(1^0\), use logarithms and L'Hôpital's rule to simplify the limit.

Answer 1

The Correct Option is C

We are required to evaluate the limit: \[ \lim_{x \to 1} x^{(1-x)} \] This is an indeterminate form of the type \(1^0\), so we can apply logarithms to simplify: \[ y = x^{(1-x)} \quad \Rightarrow \quad \ln(y) = (1-x) \ln(x) \] As \(x \to 1\), we observe that \(\ln(x) \to 0\) and \(1 - x \to 0\), so applying L'Hôpital's rule to the limit of \(\frac{\ln(x)}{1-x}\) gives: \[ \lim_{x \to 1} \frac{\ln(x)}{1-x} = -1 \] Thus, \(\ln(y) = -1\), so \(y = e^{-1} = \frac{1}{e}\).