After taking the first derivative, I still had 0 in the denominator. I then took the second and third derivatives and got the right answer. But in class, our teacher only talked about using first derivative.
So is it okay? Help!
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6$\begingroup$ Yes. ${}{}{}{}$ $\endgroup$– user296602Commented Jan 5, 2016 at 8:56
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1$\begingroup$ $\dfrac{d^2y}{dx^2}$ is the first derivative of $\dfrac{dy}{dx}$ (with respect to $x$). $\endgroup$– BumblebeeCommented Jan 5, 2016 at 9:16
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2$\begingroup$ You know all you need to decide alone ! L'Hospital transforms a limit into an equivalent one. As long as the conditions of validity are fulfilled, you may iterate. $\endgroup$– user65203Commented Jan 5, 2016 at 9:39
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3 Answers
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Only if both the limits in numerator and denominator are zero.
If only the denominator is zero, then the overall limit does not exist.
If you are forced to go to the third derivatives, it would be best to check the result using truncated Taylor series, in many training examples this actually gives a faster and clearer solution.
answered Jan 5, 2016 at 8:59
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One statement of L'Hôpital says
If $f$ and $g$ are differentiable in a neighborhood of $a$ and $f(a)=g(a)=0$, then $$ \lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)} $$ assuming the limit on the right exists.
Substituting $f\mapsto f'$ and $g\mapsto g'$ and then $f\mapsto f''$ and $g\mapsto g''$ into the statement above gives the extension required to support the ability to apply L'Hôpital multiple times.
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If what you call by "second" and "third" satisfy the initial conditions of L'H Rule, you can use L'H successively, there is nothing wrong with it.
