Does L'Hopitals Rule hold for second derivative, third derivative, etc…? Assuming the function is differential up to the $k$th derivative, is the following true?
$$\lim_{x\to c} \frac{f(x)}{g(x)} = \lim_{x\to c} \frac{f'(x)}{g'(x)} = \lim_{x\to c} \frac{f''(x)}{g''(x)} = \cdots = \lim_{x\to c} \frac{f^{(k)}(x)}{g^{(k)}(x)} = \lim_{x\to c} \frac{f^{(k+1)}(x)}{g^{(k+1)}(x)}$$
Best Answer
When L'Hopital's rule is stated by saying that $\displaystyle \lim_{x\to c} \frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)},$ one of the hypotheses is that $\lim_{x\to c}f(x) = \lim_{x\to c} g(x) = 0,$ or else both limits are among the two objects $\pm\infty.$ In order to take it one more step and say $\displaystyle \lim_{x\to c} \frac{f'(x)}{g'(x)} = \lim_{x\to c}\frac{f''(x)}{g''(x)},$ you would need to know that $\lim_{x\to c} f'(x) = \lim_{x\to c} g'(x) = 0$ or else both of those are among $\pm\infty.$ And you don't need to know anything beyond L'Hopital's rule as stated at the outset in order to know that.
Another hypothesis of L'Hopital's rule is that the limit to the right of the "equals" sign exists. You have no guarantee that L'Hopital's rule can be applied until you know that.
(Where you wrote $n\to c,$ I wrote $x\to c$.)