Does there exist a proof for L’Hôspital’s rule without relying on the mean value theorem?
Or do all proofs essentially rely on MVT?
2026-04-28 13:47:49.1777384069
Prove L’Hôspital’s rule without the mean value theorem
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There is a special case of L'Hôpital’s theorem. We assume that $f(c) = g(c) = 0$, $f$ and $g$ are differentiable at $c$, and have continuous derivatives. Then
$$\color{blue}{\lim_{x\to c} \frac{f(x)}{g(x)}} = \lim_{x \to 0} \frac{f(x+c) - f(c)}{g(x+c) - g(c)} = \frac{\lim_{x \to 0} \frac{f(x+c) - f(c)}{x}}{\lim_{x \to 0} \frac{g(x+c) - g(c)}{x}} = \frac{f'(c)}{g'(c)} = \color{blue}{\lim_{x\to c} \frac{f'(x)}{g'(x)}}$$.