I'm trying to solve $\lim_\limits{x\to0}{{\frac{x^2-7x+4}{x-3}}^{\frac{x+1}{x-7}}}$ and I'm given the following "property": $$\lim_\limits{x\to a}{[f(x)]^{g(x)}}=e^{\lim_\limits{x\to a}{[f(x)-1]·[g(x)]}}$$ With $f(x)>0$, $\lim_\limits{x\to a}{f(x)=1}$ and $\lim_\limits{x\to a}{g(x)=\infty}$
I can solve the limit with the property just fine. But I don't understand where it comes from, and I can't find anything like it by searching on Google or here, can someone shed a light on this? Does this "special limit" have a name? or more formal restrictions?
For $x$ near $0$ the function $\frac {x^{2}-7x+4} {x-3}$ takes negative values so the function is not a well defined real valued function near $x=0$.
The formula in the title is not always true.