Behavior of $\sin x/x$ as $x$ approaches 0? Which is the limiting behavior, $x \to 0\ \frac{\sin x}{x}$, in terms of x:
1) $\lim\limits_{x \to 0} \frac{\sin x}{x}$ = $\lim\limits_{x \to 0}\frac{\sin'x}{x'} = \frac{\lim\limits_{x \to 0}\cos x}{1} \rightarrow 1 - x^2/2 + O(x^4)$ by L'Hospitals' rule, or,
2) As $x \to 0,\  \frac{\sin x}{x}\sim\frac{x - x^3/3! + O(x^5)}{ x } = 1 - x^2/6 + O(x^4)$ expanding $\sin x$ by Taylor series.
I know each has a limit of 1, but what is the behavior in terms of the small $x$, and why don't both approaches give the same answer?

!) is incorrect as stated by another person .
2)is correct provided you remove all the limit signs in the equations . The limit sign when added makes the expression denote just the  number which is the expressed limit. 
