calculus - Avoiding circular logic using L'Hospital's rule

Often, using L'Hospital's rule can make a limit much simpler to evaluate, but in some circumstances it can be incorrect to use the rule even when all of its criteria are met - one example being the evaluation of $\lim_{x\to0}\frac{\sin x}{x}$, which relies on $\frac{d}{dx}\sin x=\cos x$ being known, which itself relies on the limit we are trying to prove!

How, when considering using L'Hospital's rule, can examples like this one be spotted in order to avoid circular logic?

At first glance, I wouldn't have known that the proof was circular, and I'm concerned that I might make similar mistakes with other functions.

Answer

Use of the rule can never be incorrect in the situations you're describing.

One may argue that in some cases doesn't tell you anything you didn't already know, but that does not mean that the conclusions you reach from it are in any danger of being false.

At worst you can say that using L'Hospital's rule on $\lim_{x\to 0} \frac{\sin x}{x}$ is a detour compared to recognizing the original limit as being the definition of $\sin'(x)$ at $0$ -- but that doesn't put the truth of the result at risk. The limit IS the derivative of the sine, no matter whether you reach this conclusion by L'Hospital or by pattern-matching the definition of a derivative.

If you have a valid way of finding the derivative other than applying the definition directly (and this will usually be the case; it is extremely rare to need to calculate derivatives from first principles rather than symbolically), then it doesn't matter how you discovered that this derivative is what you're looking for.

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Of course if what you're doing is learning for the first time what the derivative of the sine function is, then L'Hospital will not help you. This is not because it is not valid, but because what you can conclude then is at most that what you're looking for is the thing you were looking for -- which is true but useless. (And even that depends on actually knowing that $\sin$ is continuously differentiable in the fist place).

If at that point, in that specific context you decide to proceed by "and we know the derivative of sine is cosine", then you will be guilty of circular reasoning. But the error is then not that you used L'Hospital, but that external to your use of L'Hospital you decided to assume something you hadn't actually finished proving yet.