L’Hôpital’s rule practice question 25

The bottom is $x^{4}$: how many times must you differentiate it before it stops vanishing at 0? Keep track of the factor each derivative of $\cos(2x)$ brings out.

Every application leaves a $\tfrac{0}{0}$ form until the fourth. Keep the constants straight: the $n$th derivative of $\cos(2x)$ carries a factor $2^{n}$.

After two applications the quotient is $\dfrac{4 - 4\cos(2x)}{12x^{2}}$, still $\tfrac{0}{0}$. Two more applications finish it; cancel common factors between steps to keep the numbers small.