\[\dfrac{\sin^2x -x^2\cos^2x}{x^4\cos^2x}\]\[=\dfrac{(x-\dfrac16 x^3 +O(x^5))^2-x^2(1-\dfrac12 x^2+O(x^4))^2}{x^4(1+O(x^2))}\]\[= \dfrac{x^2-\dfrac13 x^4 +O(x^6)-x^2+ x^4+O(x^6)}{x^4+O(x^6)}\]\[= \dfrac{\dfrac23 x^4 +O(x^6)}{x^4+O(x^6)}= \dfrac{\dfrac23 +O(x^2)}{1+O(x^2)} \rightarrow \dfrac23.\]