I am trying to solve the following limit:
$ \lim_{x \to 1^+} \frac{x^x+\frac{x}{2}-\sqrt x-\frac{1}{2}}{\log(x)-x+1} $
My attempt was to substitute $x$ with $1+y$, which results in the following:
$ \lim_{y \to 0^+}_{y<1} \frac{(y+1)^{y+1}+\frac{y}{2}-\sqrt{1+y}}{\sum_{n=1}^\infty (-1)^n\frac{y^{n+1}}{n+1}} $
I strongly suspect that the limit does not exists (the term seems to diverge towards negative infinity), but I don't know how to proceed from here. I think that it might be easiest to find a null sequence to substitute $y$ with, that shows that the values of the term above are unbounded, but I am stuck at this for some time now.
Could somebody point me into the right direction?
UPDATE: Unfortunately I am not allowed to use De L'Hopital's rule yet.