Calculus Derivative—Finding unknown constants 
Find all values of $k$ and $l$ such that
  $$\lim_{x \to 0} \frac{k+\cos(lx)}{x^2}=-4.$$

Any help on how to do this would be greatly appreciated.

(Without involving big O notation). The limit of denominator goes to zero  and numerotor goes to $k+1$ as $x\rightarrow 0$. Since the limit exists, $k+1=0$ and hence $k=-1$. Then applying L-H rule ( since we have $\frac{0}{0}$ indeterminate form) we get $\lim_{x\rightarrow 0}\frac{-\ell\sin(\ell x)}{2x}$ which is equal to $\frac{-\ell^2}{2}$. Therefore, $-\frac{\ell^2}{2}=-4\Rightarrow \ell=2\sqrt{2}$ or $\ell=-2\sqrt{2}$. 
