limits - Is L'Hopital for $limlimits_{xto0}frac{sin(x)}{x}$ circular?

I was considering using L'Hopital for $\displaystyle\lim\limits_{x\to0}\frac{\sin(x)}{x}$, but I was told that this is circular, because we use this limit to show $\displaystyle\frac{\mathrm d}{\mathrm dx}\sin(x) = \cos(x)$.

Do we have to use this limit to find the derivative of $\sin(x)$, or is there a legitimate counter-argument here?