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I'm familiar to both the $O$ and little $o$ notation. I know they are of great use when studying limits, asymptotics and series. However, I'm not sure when to use one over another, particularly when solving problems. For example, you can put that, for $x \to 0$

$\sin x = x + o(x)$

since

$\frac{\sin x}{x}-1 \to 0$

I've also recently read about a great definition for the derivative and strong derivative (Knuth), respectively, for $\epsilon \to 0$:

f(x+\epsilon)=f(x)+\epsilon f'(x)+o(\epsilon)

which clearly means that

\frac{f(x+\epsilon)-f(x)}{\epsilon}- f'(x) \to 0

and the same for

f(x+\epsilon)=f(x)+\epsilon f'(x)+o(\epsilon ^2)

Now consider the following

$\lim_{x \rightarrow 0}{\frac{\int_{0}^{x} \frac{\sin(t)}{t}-\tan(x)}{2x(1-\cos(x))}}$

I know that

$2x\left( {1 - \cos x} \right) = {x^3} + o\left( {{x^3}} \right)$

$\tan x = x + \frac{x^3}{3} + o\left( x^3 \right)$

$\int\limits_0^x {\frac{{\sin t}}{t}dt = x -\frac{x^3}{3\cdot 3!} +o\left( x^3 \right)} $

So this gives

$\frac{{x - \frac{{{x^3}}}{{3 \cdot 3!}} + o\left( {{x^3}} \right) - x - \frac{{{x^3}}}{3} - o\left( {{x^3}} \right)}}{{{x^3} + o\left( {{x^3}} \right)}} = $

$ \frac{{\left( { - \frac{1}{3} - \frac{1}{{18}}} \right){x^3} + o\left( {{x^3}} \right)}}{{{x^3} + o\left( {{x^3}} \right)}}$

$\frac{{\left( { - \frac{7}{{18}}} \right){x^3} + o\left( {{x^3}} \right)}}{{{x^3} + o\left( {{x^3}} \right)}} = - \frac{7}{{18}}$

So how could I use the big $O$ to solve the same problem? And why should I choose big $O$ over little $o$, when $o$ seems more accurate than $O$?

NOTE: When I talk about the big $O$ I'm considering it for any $x \to a$, not only for asymptotic behaviour $x \to \infty$. For example, for $x \to 0$

$\sin x = O(x) $

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    It is generally better to use big $O$ notation than little $o$ because big $O$ gives rise to an equivalence relation among functions whereas little $o$ does not.2015-11-12

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Let's look at $\sin x$ as $x\to0$. You can write $\sin x=O(x)$, or $\sin x=x+o(x^2)$, or $\sin x=x+O(x^3)$, or $\sin x=x-(1/3)x^3+o(x^4)$, or .... You use whichever one works out nicest in whatever problem it is that you are solving.