L’Hôpital’s rule practice question 24

What do $\tan x - x$ and $x - \sin x$ each tend to at 0? One round of the rule may not settle it, so recheck the form after each round.

After one application the quotient is $\dfrac{\sec^{2}x - 1}{1 - \cos x}$, which is still $\tfrac{0}{0}$.

Either apply the rule again (the derivative of $\sec^{2}x$ is $2\sec^{2}x\tan x$), or write $\sec^{2}x - 1 = \dfrac{1 - \cos^{2}x}{\cos^{2}x}$ and cancel the factor $1 - \cos x$.