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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?

$$\lim_{x\to c} \frac{f(x)}{g(x)} = \lim_{x\to c} \frac{f'(x)}{g'(x)} = \lim_{x\to c} \frac{f''(x)}{g''(x)} = \cdots = \lim_{x\to c} \frac{f^{(k)}(x)}{g^{(k)}(x)} = \lim_{x\to c} \frac{f^{(k+1)}(x)}{g^{(k+1)}(x)}$$

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    $\begingroup$ If $f'$ and $g'$, etc. satisfy the hypotheses then yes. $\endgroup$ – NeedForHelp Feb 9 '17 at 1:11
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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 $x\to c$.)

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  • $\begingroup$ Actually, in the case the $g\to \infty$, then LRH is still applicable, even if $\lim f$ fails to exist. $\endgroup$ – Mark Viola Feb 9 '17 at 1:21
  • $\begingroup$ @Dr.MV :-/ That's that weird case I'd personally just ignore. Why would you even apply L'Hospital's rule in that case? $\endgroup$ – Simply Beautiful Art Feb 9 '17 at 1:23
  • $\begingroup$ @simplybeautifulart In the case for which $f$ approaches a finite limit, it's applicable but unnecessary. But if the numerator does not approach a finite limit, then LHR is applicable and might be useful. $\endgroup$ – Mark Viola Feb 9 '17 at 1:44
  • $\begingroup$ @Dr.MV Well that is almost too obvious. :-| $\endgroup$ – Simply Beautiful Art Feb 9 '17 at 1:45
  • $\begingroup$ @Dr.MV : Can you cite an instance of the application of the rule in that situation? $\endgroup$ – Michael Hardy Feb 9 '17 at 2:47
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Well, let's see. Under the specified conditions and $f(c)=g(c)=0$ (or both go to $\pm\infty$), then

$$\lim_{x\to c}\frac{f(x)}{g(x)}=\lim_{x\to c}\frac{f'(x)}{g'(x)}$$

Now, if $f'(x)$ and $g'(x)$ still hold all those conditions and $f'(c)=g'(c)=0$ (or both go to $\pm\infty$), then

$$\lim_{x\to c}\frac{f'(x)}{g'(x)}=\lim_{x\to c}\frac{f''(x)}{g''(x)}$$

And you may repeat this process until the limit is easily evaluated or it fails to uphold the conditions of L'Hospital's rule.

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