Can x*log(x) Be Approximated As x Approaches 0?

[jostpuur]

Do there exist numbers $A,B,C$ such that

$$x\log(x) = Ax^B + O(x^C)\quad\quad\textrm{as}\; x\to 0^+$$

and such that

$$1\leq C$$

?

The approximation is trivial if $C < 1$, because then $x^{1-C}\log(x)$ would approach zero, and $A$ and $B$ could be chosen to be almost anything (only $C<B$ needed). But if $1\leq C$, then the approximation could have some content. Obviously conditions

$$A < 0 < B < 1$$

should hold, because $x\log(x) < 0$ when $0<x<1$, and $D_x(x\log(x))\to \infty$.

update:

I see these numbers do not exist, because if they did, then also $\log(x)$ could be approximated with some $\alpha x^{\beta}$ where $\beta <0$.

 

[fresh_42]

I would work with series expansions. Taylor e.g. has a lot of approximations of the remainder.