L'Hospital's rule's hypothesis that the right hand limit should exist Why does the l'Hospital's rule assume that the right hand limit should exist? How does it work for x ln(-x) as x tends to 0 from the the left hand side?

For $x\ln(-x)$, we can re-write this as $$\frac{\ln(-x)}{\frac{1}{x}}$$
Now L'Hospitals rule actually states that if we take the derivative of the top and bottom and get a finite number (or $\pm\infty$), then the limit is the same as the original limit.
However, if we take the derivatives, we get
$$\frac{\frac{1}{x}}{\frac{-1}{x^2}}$$
and as $x$ goes to zero, this limit is clearly undefined. Thus L'Hopital's rule does not apply here.
Reference here.
