The taylor's series expansion of f( x)in powers of (x+a) is
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A Taylor expansion of f around a allows to simply read off the the derivatives of arbitrary order at a. So its just a neat way to write the function which includes much information how this function behaves near that point.
This is useful for example when calculating limits of functions when x→a (l'Hospital rule).
As an example, if you have the polyomials f(x)=2(x−1)+3(x−1)2 and g(x)=x−1, then the limit
limx→1
f(x)
g(x)
= limx→1
2(x−1)+3(x−1)2
x−1
=2,
is easy to see. However, if we start with f(x)=3x2−4x+1, then calculating the limit is (still easy as the functions are easy, but) not trivial anymore.
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