Accusation of circular reasoning in finding $\lim\limits_{x \to 0} \frac{\sin x}{x}$ $$\lim\limits_{x \to 0} \frac{\sin x}{x}= \cos 0 = 1$$
This result follows from either L'Hospital's rule or by definition of derivative of $\sin x$ at $x=0$. I see widespread accusation that finding the limit this way is circular reasoning. However, I can hardly agree with it. Otherwise, you may as well accuse that summing an Infinite Series by Definite Integral is circular.
In my humble opinion, relating a limit back to the derivative or integral of a function should be a perfectly legitimate (and sometimes very clever) trick. Why is it accused as circular reasoning by so many people?
Furthermore, it seems that a majority of people making this accusation fail to provide a better way to find the limit concerned. Many of their attempts assume the area of a sector to be known. I don't see how assuming the area of sector to be $x/2$ is any better than assuming the derivative of $\sin x$ to be $\cos x$.
Feel free to share your point of view on this.
Below are some discussions on the limit concerned.
https://www.maa.org/sites/default/files/pdf/mathdl/CMJ/Richman160-162.pdf
http://forums.xkcd.com/viewtopic.php?t=112236
 A: I once saw the definition of $\sin(x)$ for $x\in [-\frac{\pi}{2}, \frac{\pi}{2}]$ as the inverse function of
$$\arcsin(t) = \int_0^t\frac{1}{\sqrt{1-u^2}} du$$
It follows that
$$\sin^\prime(x) = \sqrt{1-\sin^2(x)}$$
hence $\sin^\prime(0) = 1$ and it follows that $\lim_\limits{x \to 0} \frac{\sin(x)}{x} = 1$.
A: Normally this result is needed to derive the derivative of trigonometric functions (with the geometric interpretation) in the first place. So using the derivative for the limit makes this seem circular from this point. 
On the other hand if you use the taylor series as the definition of trig functions then the limit is trivial by using the series of the sine function. And in order to derive the taylor series we also first need the derivatives of sine and cosine. Hence, one could argue that this is again circular.
The geometric way of deriving the limit has also flaws but it uses very simple concept that is why I would say that it is better than using the derivative/taylor series derivation.
The only thing you have to accept for the geometric derivation is this inequality
$$\sin x\leq x \leq \tan x $$ 
for small positive $x$ this is very easy to see from the geometric picture of a unit circle.
