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.

Use polynomials long division to have the form
$$ f(x) = ax^3+bx^2+cx+d=(( 3a-b+c )x-2a+2b+d)(x^2+x-2)$$

$$ \implies f(x) = (( 3a-b+c )x-2a+2b+d) g(x) $$

which simplifies evaluating the limit and the problem becomes straightforward.
