HMMT 2015 A3

Let $p$ be a real number and $c \neq 0$ an integer such that

$$c-0.1<x^{p}\left(\frac{1-(1+x)^{10}}{1+(1+x)^{10}}\right)<c+0.1$$

for all (positive) real numbers $x$ with $0<x<10^{-100}$. (The exact value $10^{-100}$ is not important. You could replace it with any "sufficiently small number".) Find the ordered pair $(p, c)$.