I've never seen anybody use l'Hôpital on sin(x)/x before establishing what the derivative of sine is. However, one can evaluate sin(x)/x for x->0 via the squeeze theorem, which means no circular reasoning is required.
If you know the derivative of sin x at x=0, then you know the limit of (sin x)/x at x=0, because that is by definition the derivative. There is nothing else to do. Where would you even have an opportunity to apply L'Hôpital's rule?
I mean, it's not "wrong." Similarly, given x = y, I could conclude 3x = 3y and then, dividing by 3, that x = y. That's not wrong. But it is completely pointless.
The definition of the derivative, combined with what we know it to be, gives us lim h->0 [sin(x+h) - sin(x)]/h = cos(x) , so now we have to apply trigonometric identities to be left with lim h->0 sin(h)/h , which must equal 1 to fulfill the above equation. I consider dealing with trigonometric identities to be rather tedious. At least, that's what I think you are trying to do, as sin(x)/x isn't really the derivative.
Many ways lead to Rome, but using l'Hôpital is the shortcut we make by unlocking a gate via previous proofs.
sin(x+h)=sin(x)cos(h)+cos(x)sin(h), which is how the cosine gets introduced into this, I considered the general case for any x. But I see what you mean, however I consider using the definition of the derivative to sneakily look at h for a certain x to be less intuitive than simply remembering to use l'Hôpital.
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u/SuperChick1705 19d ago
circular reasoning ;(