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 \((-1, \; -5)\)