Evaluate $\lim _{x\to 0}\frac{1-\frac{x^2}{2}-\cos(\frac{x}{1-x^2})}{x^4}$ $$\lim _{x\to 0}\frac{1-\frac{x^2}{2}-\cos(\frac{x}{1-x^2})}{x^4}$$
An approach I can think about is to expand $\cos$ using taylor series, is there another approach?

Taylor expand 
$$\frac x{1-x^2}=x+x^3+x^5+O(x^7),\>\>\>\>\>\cos t = 1-\frac{t^2}2+\frac{t^4}{24}+O(t^6)$$ 
to get
$$\cos\frac{x}{1-x^2}=1-\frac{1}2x^2-\frac{23}{24}x^4+O(x^6)$$
Thus,
$$\lim _{x\to 0}\frac{1-\frac{x^2}{2}-\cos(\frac{x}{1-x^2})}{x^4}$$
$$=\lim _{x\to 0}\frac{1-\frac{x^2}{2}-(1-\frac{x^2}2-\frac{23}{24}x^4+O(x^6))}{x^4} $$
$$=\lim _{x\to 0}\frac{\frac{23}{24}x^4+O(x^6)}{x^4}=\frac{23}{24}$$
