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I recently read the following definition:

Let $f$ be a function defined in a neighbourhood $I$ of a point $x_0$. We say that the linear function $L(x)=a(x-x_0)+f(x_0)$ approximates the function $f$ in $I$ if exists a function $\delta$ defined in $I$ such that

$f(x)=L(x)+(x-x_0)\delta(x)$ , $\forall x\in I$

and

$\lim_{x\to x_0}\delta(x)=\delta(x_0)=0$

I don't understand the sense of this definition.

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