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Saying that the polynomial P, of degree at most n, is the Taylor expansion of order n about 0 of a function f means that the quantity R(x) = f (x) ‒ P(x) is negligible compared with *x n, written R(x) = o(x n*).
This describes the function's behavior near 0—a local analysis.
For many purposes, such as finding limits, this is more than sufficient. But this formulation cannot be used to study an entire interval—in other words, to tackle a global problem.
Lagrange's remainder
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