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In the CLRS chapter 3: When $x → 0$, the approximation of $e^x$ by $1+x$ is quite good:

$$e^x = 1 + x + Θ(x^2).$$

How is it to be interpreted, what is the role of asymptotic notation here?

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There are constants $c,C,\epsilon > 0$, such that $1+x+cx^2 \le e^x \le 1+x+Cx^2$ whenever $|x| \leq \epsilon$.

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This is an example of "placeholder notation". In such expressions, you should mentally replace "$\Theta(x^2)$" with "$f$, for some function $f=\Theta(x^2)$".

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