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Today's calculation of Integral 825

Source: 2012 Hokkaido University entrance exam/Science

July 1, 2012
calculusintegrationtrigonometrylimitcalculus computations

Problem Statement

Answer the following questions.
(1) For x≥0x\geq 0, show that x−x36≤sin⁡x≤x.x-\frac{x^3}{6}\leq \sin x\leq x.
(2) For x≥0x\geq 0, show that x33−x530≤∫0xtsin⁡t dt≤x33.\frac{x^3}{3}-\frac{x^5}{30}\leq \int_0^x t\sin t\ dt\leq \frac{x^3}{3}.
(3) Find the limit
lim⁡x→0sin⁡x−xcos⁡xx3.\lim_{x\rightarrow 0} \frac{\sin x-x\cos x}{x^3}.
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