TITLE: L'Hopital's rule and $\frac{\sin x}x$
QUESTION [11 upvotes]: 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.

REPLY [7 votes]: I expand my comment to ziggurism's answer here.

Suppose that by a lot of hardwork, patience and ... (add more nice words if you like) I have obtained these three facts:

$\cos x$ is continuous.
$\dfrac{d}{dx}(\sin x) = \cos x$
L'Hospital's Rule

and my next goal is establish the following fact: $$\lim_{x \to 0}\frac{\sin x}{x} = 1\tag{*}$$ Because of all that hardwork, patience and ... I know that my goal $(*)$ is an immediate consequence of just the fact $(2)$ alone stated above. Then why would I combine all the three facts mentioned above to achieve my goal? To borrow an idea from user "zhw.", wouldn't a person doing this would be considered silly?
More often than not, many students don't really understand what's going behind the scenes when we use the mantra of "differentiate and plug" championed by L'Hospital's Rule. The act of differentiation itself entails that we know certain limits (and rules of differentiation) and further most of the derivatives are continuous (so that plugging works after differentiation step).
If one is so fond of L'Hospital's Rule why not put that to a better use to solve complex problems (like this and this) instead of using it to obtain limits which are immediate consequences of differentiation formulas.