Could the indeterminate form limit $\lim_{x\to a} [\frac{x^m-a^m}{x^n-a^n}]=\frac{m}{n}a^{m-n}$ be extended to all real numbers?

Question

In my text book I found that:

$$\lim_{x\to a} \left[\frac{x^m-a^m}{x^n-a^n}\right]=\frac{m}{n}a^{m-n}$$ where $m$ and $n$ are any positive integers.

But could this be extended to any real number including $\ln{5}$ for example ?