Limit of an unknown cubic expression Let $f(x)=ax^3+bx^2+cx+d$ and $g(x)=x^2+x-2$. If $$\lim_{x \to 1}\frac{f(x)}{g(x)}=1$$ and $$\lim_{x \to -2}\frac{f(x)}{g(x)}=4$$ then find the value of $$\frac{c^2+d^2}{a^2+b^2}$$ Since the denominator is tending to $0$ in both cases, the numerator should also tend to $0$, in order to get indeterminate form. But it led to more and more equations.

What about? $$f(x)=-x^3+x^2+4x-4$$
So:
$$\frac{c^2+d^2}{a^2+b^2}=32/2=16$$

Let us make ansatz that $$f(x)=a(x+\alpha)(x-1)(x+2)$$
So:
$$a(1+\alpha)=1,a(\alpha-2)=4\implies a=-1,\alpha=-2$$
so:
$$f(x)=-(x-1)(x+2)(x-2)=-x^3+x^2+4x-4$$
