2. Let

$$f(x)=\frac{x^2}{\sqrt{1+x^2}}-x^2$$ $$g(x)=\cos x\sin(x^2)-x^2.$$

i) $(10\%)$ Compute the coefficients of $x^4$ and $x^9$ in the Taylor series of $f(x)$ at $x=0$.

ii) $(10\%)$ Compute the coefficients of $x^4$ and $x^6$ in the Taylor series of $g(x)$ at $x=0$.

iii) $(10\%)$ Compute $\lim_{x\to 0}\dfrac{f(x)}{g(x)}$.