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The Lagrange error bound gives a precise way to measure how accurate a Taylor polynomial approximation really is, and for a function like sin(x), it reveals something powerful about how quickly the approximation improves. When you approximate sin(x) using its Taylor polynomial centered at a point (usually 0), you are essentially replacing the function with a finite sum of polynomial terms. The natural question is: how far off is that approximation from the true value? The Lagrange error bound an
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