HMMT 2015 Algebra #3

Let P be a real number and $c \ne 0$ an integer such that$c-0.1 < x^p \left ( \dfrac{1-(1+x)^10}{1+(1+x)^10} \right ) < c+0.1$ 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 $(p, \;c)$. 

정답 $(-1, \; -5)$