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Q26Section A (Section A)26 of 35

Limit of a ratio vanishing to third order at one half

The value of the limit

lim⁡x→12∫1/2xcos⁡2πt dte2x2−e(x2+14)\lim_{x\to\frac12}\frac{\int_{1/2}^x\cos^2\pi t\,dt}{\frac{e^{2x}}2-e\left(x^2+\frac14\right)}

is

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