\lim_{x\to 0}\frac{\log(\cos x)}{(1+x^2)^{11}-1}=\lim_{x\to 0}\left(\frac{\log(\cos x)}{\cos x-1}\cdot \frac{\cos x-1}{x^2} \cdot \left(\frac{(1+x^2)^{11}-1}{x^2}}\right)^{-1}\right)=1\cdot \left(-\frac{1}{2}\right) \cdot \frac{1}{11}=-\frac{1}{22}