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Calculus, Taylor series Consider the function f(x) = sin(x) x . 1. Compute limx→0 f(x) using...

Calculus, Taylor series Consider the function f(x) = sin(x) x . 1. Compute limx→0 f(x) using l’Hˆopital’s rule. 2. Use Taylor’s remainder theorem to get the same result: (a) Write down P1(x), the first-order Taylor polynomial for sin(x) centered at a = 0. (b) Write down an upper bound on the absolute value of the remainder R1(x) = sin(x) − P1(x), using your knowledge about the derivatives of sin(x). (c) Express f(x) as f(x) = P1(x) x + R1(x) x , and compute the limits of the two terms as x → 0.

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