In our material we have the statement that for a given $x\in\mathbb{R}$ with $x\geq 1$ the function $f(t):[0,1]\to \mathbb{R}$, where $f(t)=t^{x-1}$, is continuous (if we define $0^0:=1$).
Then a few lines later we assume $x\in\mathbb{R}$ and $x\in]0,1[$ and consider another function $g:]0,1[\to\mathbb{R}$ with $g(\alpha)=\alpha^x$. Here $]0,1[$ is defined as the open interval from $1$ to $0$. We compute the limit: $\lim\limits_{\alpha\to 0}g(\alpha)= \lim\limits_{\alpha\to 0}\alpha^x$. However, in this case we suddenly rewrite $\lim\limits_{\alpha\to 0}\alpha^x=\lim\limits_{\alpha\to 0}e^{x\ln(\alpha)}$ in order to check the limit.
Why did we not use the previous result that we already know that $f(t)=t^{x-1}$ is continous? With this we could have simply said: $\lim\limits_{\alpha\to 0}\alpha^x=(\lim\limits_{\alpha\to 0}\alpha)^x=0^x=0$. Am I wrong?
The short answer is that when you change domains you change the function. Since you now have a left side to approach zero from you must ensure it also converges to the same number, otherwise it would be discontinuous there. For example if we take a step function and restrict the domain to one of the steps it is constant, and therefore continuous on that domain however it is certainly not continuous everywhere being a step function. This means you can't use the previous result because it doesn't apply to the new function in the larger domain.