Can all limits be solved without L'Hopital?

Question

Is there any type of limit that requires L'Hopital's rule to solve or can all limits be solved without using it?

If all limits can be solved without L'hopitals, is there some sort of proof or intuition for this?

Answer 1

Usually Taylor's expansion has the same strength. However, if you prohibit both L'Hopital and Taylor, then I doubt that $\lim_{x\to 0} \frac{\sin x-x}{x^3}$ or $\lim_{x\to 0} \frac{\sin x-x+x^3/6}{x^5}$ can be solved elementary, without using the ideas of L'hopital or Taylor in some sense.

Answer 2

If you cannot use L'Hopital's rule you can always use Cauchy's mean-value theorem together with the squeeze theorem.