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Suppose that, as $x$ vanishes, the function $f(x)$ is approximated by the Laurent series $$f(x) \approx \frac 1 x - i\pi - \frac x 4 + \frac{i\pi x^2}{12} + \cdots$$

Question 1: can I write the following? $$\lim_{x\rightarrow 0} f(x) = \frac 1 x - i\pi$$ Or is this style unconventional?

Question 2: what if, later in the same paper, having discovered an interest in the $x^1$-order term, I write the following?

$$\lim_{x\rightarrow 0} f(x) = \frac 1 x - i\pi - \frac x 4$$

I suspect that my style is imperfect. Is there a more standard style in which to write what I mean to say?

I should not solicit opinions on this site, so am asking about conventions and standards rather than about preferences. If you know of more than one convention or standard, then please feel free to mention more than one.

For information, my audience are engineers, which means that they'll probably tolerate my misstyling in any case, but I would rather not be more mathematically uncouth that I can help. This is why I ask.

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    $\begingroup$ I would write $f(x)=1/x-i\pi+o(1)$ and say in prose that I was truncating a series and that $o(\cdot)$ was to be understood as $x\to0$. $\endgroup$ – kimchi lover May 22 at 21:03
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    $\begingroup$ I think $f(x) = 1/x - i\pi +O(x)$ is better notation. $\endgroup$ – Somos May 22 at 22:12

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