I have this limit,
$$ \lim_{b\rightarrow0}b\log b +(1-cb-b)\log(1-cb-b)-(1-cb)\log(1-cb) $$
where $c\in\mathbb{R}$ is a constant.
I know this is going to zero, but I want to know how fast it decreases. For instance, when $c=0$ it becomes
$$ \lim_{b\rightarrow0}b\log b+(1-b)\log(1-b) $$ and here the dominat term is $b\log b$ so it decreases as fast as $b\log b$. So, the dominant term in the case of the first limit should be bounded by $b\log b$ but how I get it, and what if $c$ is not constant anymore, if $c=\frac{1}{b}$, the limit still would vanish, but how fast in this case?
thanks
We have that
$$b\log b +(1-cb-b)\log(1-cb-b)-(1-cb)log(1-cb)=$$
$$=b\log b+(1-cb-b)(-cb-b+o(b))-(1-cb)(-cb+o(b))=$$$$=b\log b-cb-b+cb+o(b)=b\log b-b+o(b)=$$
$$=b\log b + o(b\log b)$$