We are essentially studying the rational function $f(x):=\frac{1-(1+x)^{10}}{1+(1+x)^{10}}=\frac{-10 x+O\left(x^{2}\right)}{2+O(x)}$. Intuitively, $f(x) \approx \frac{-10 x}{2}=-5 x$ for "small nonzero $x$ ". So $g(x):= x^{p} f(x) \approx-5 x^{p+1}$ for "small nonzero $x$ ". If $p+1=0, g \approx-5$ becomes approximately constant as $x \rightarrow 0$. Since $c$ is an integer, we must have $c=-5$ (as -5 is the only integer within 0.1 of -5 ).