The expression \[\mathop {\lim }\limits_{\alpha \to 0} \frac{{{{\text{x}}^\alpha } – 1}}{\alpha }\] is equal to A. log x B. 0 C. x log x D. \[\infty \]

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 $\infty$.