- #1
jostpuur
- 2,112
- 19
Do there exist numbers A,B,C such that
<br /> x\log(x) = Ax^B + O(x^C)\quad\quad\textrm{as}\; x\to 0^+<br />
and such that
<br /> 1\leq C<br />
?
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
<br /> A < 0 < B < 1<br />
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.
<br /> x\log(x) = Ax^B + O(x^C)\quad\quad\textrm{as}\; x\to 0^+<br />
and such that
<br /> 1\leq C<br />
?
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
<br /> A < 0 < B < 1<br />
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.
Last edited:
