The correct answer is A. log x.
Let’s take a look at each option in turn.
Option A: log x. This is the correct answer. As $\alpha$ approaches 0, $x^\alpha$ approaches 1, so the expression $\frac{{x^\alpha } – 1}}{\alpha }$ approaches $\log x$.
Option B: 0. This is not the correct answer. As $\alpha$ approaches 0, $x^\alpha$ approaches 1, so the expression $\frac{{x^\alpha } – 1}}{\alpha }$ approaches $\log x$, which is not equal to 0.
Option C: x log x. This is not the correct answer. As $\alpha$ approaches 0, $x^\alpha$ approaches 1, so the expression $\frac{{x^\alpha } – 1}}{\alpha }$ approaches $\log x$, which is not equal to $x \log x$.
Option D: $\infty$. This is not the correct answer. As $\alpha$ approaches 0, $x^\alpha$ approaches 1, so the expression $\frac{{x^\alpha } – 1}}{\alpha }$ approaches $\log x$, which is not equal to $\infty$.