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I can't compute a Taylor series of a function like $f(x)=\sqrt{x}$ to some order around $x_0=0$, because the derivative at that point doesn't exist.

If I consider the taylor series $Tf$ at any positive point, $x_0=2.7$ say, then the value at that point is necessarily exact, $Tf(2.7)=1.643...=\sqrt{2.7}$, but $Tf(0)\ne 0=\sqrt{0}$.

How can I find an aproximation, which satisfies the property that it takes the "right" values at problematic points, as motivated in the example above?

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You might want to look into multipoint Padé approximants... –  J. M. Dec 7 '11 at 22:17

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