Consider the function $f(x)=e^{-\frac{1}{x^2}}$, for all $x\
e 0\in \mathbb R$, and $f(0)=0$ (or take any other function with the property that all derivatives at $0$ vanish, but the function is not locally constant at $0$). Now suppose you are asked to compute $\lim_{x\to 0}\frac{f(x)}{f(x)}$. Of course, this limit is $1$ by simply working out the fraction first, and then taking the limit. But if you try to use L'Hopitals' rule you find that the conditions are met, but $\lim_{x\to 0}\frac{f'(x)}{f'(x)}$ is still of indeterminate form. Again L'Hopitals is applicable, and again $\lim_{x\to 0}\frac{f''(x)}{f''(x)}$ is indeterminate. This will go on forever. So even though the limit can be determined, and even though the conditions for L'Hopitals rule are repeatedly met, you will never get the result this way.