I have heard people say that you can't (or shouldn't) use the L'Hopital's rule to calculate $\lim\limits_{x\to 0}\frac{\sin x}x=\lim\limits_{x\to 0}\cos x=1$, because the result $\frac d{dx}\sin x=\cos x$ is derived by using that limit.
But is that opinion justified? Why should I be vary of applying L'Hopital's rule to that limit?
I don't see any problem with it. The sine function fulfills the conditions of the L'Hopital's rule.
Also, it is a fact that the derivative of sine is cosine, no matter how we proved it. Certainly there is a way to prove $\frac d{dx}\sin x=\cos x$ without using the said limit (if someone knows how, they can post it) so we don't even have any circular logic. Even if there isn't, $\frac d{dx}\sin x=\cos x$ was proven sometime without referencing the L'Hopital's rule so we know it is true. Why wouldn't we then freely apply the L'Hopital's rule to $\frac {\sin x}x$?
PS I'm not saying that this is the best method to derive the limit or anything, but that I don't understand why it is so frowned upon and often considered invalid.
Best Answer
It is a correct application of l'Hôpital's rule, but using it to prove that the derivative of $\sin x$ is $\cos x$ is possibly circular logic, depending on your definition of $\sin x$.
If we did not know what the derivative of $\sin x$ was, but we did know the angle sum formula, here are some steps we could take to compute that derivative, starting from first principles, i.e. the definition of the derivative.
$$ (\sin x)'= \lim_{h\to 0}\dfrac{\sin(x+h)-\sin x}{h} = \lim_{h\to 0}\dfrac{\sin x\cos h+\cos x\sin h-\sin x}{h} \\ = \cos x\cdot\left(\lim_{h\to 0}\dfrac{\sin h}{h}\right)+ \sin x\cdot\left(\lim_{h\to 0}\dfrac{\cos h -1}{h}\right) $$
To complete the computation of the derivative of $\sin x$, we must first know the limit of $\sin h/h$ and $(\cos h-1)/h$. We can't use l'Hôpital at this point because to use l'Hôpital you need to know the derivative of $\sin x$ at $0$, which is the very thing we are trying to compute. Assuming what you're trying to prove is a logical error known as "begging the question". Mathematics is axiomatic, so that every result builds on previous results, and no arguments are circular.
But if we can confirm the derivative of $\sin x$ by some other means, for example by defining it in terms of its Taylor series or a differential equation, then we may use it in a l'Hôpital computation to derive $\lim \sin x/x$ without fear of being circular. While not logically incorrect, this would still be fairly redundant: if you know the derivative of $\sin x$ just observe that, by definition, $\lim_{h\to 0}\sin x/x$ is just the derivative at 0. Using the machinery of l'Hôpital is overkill to get an answer you already know.
However if you're in a situation where you don't care about logical foundations and rigor, you're not proving theorems from first principles about calculus of trigonometric functions, you just need to compute $\lim\sin x/x$ and want to allow all known formulas and techniques, feel free to use l'Hôpital. It is correct.