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Prove some properties of improper integrals involving 1/x and sin x

  1. Prove the following limit formulas:

        \[ \lim_{h \to 0^+} \left( \int_{-1}^{-h} \frac{dx}{x} + \int_h^1 \frac{dx}{x} \right) = 0, \qquad \lim_{h \to +\infty} \int_{-h}^h \sin x \, dx = 0. \]

  2. Determine whether the following improper integrals converge:

        \[ \int_{-1}^1 \frac{dx}{x}; \qquad \int_{-\infty}^{\infty} \sin x \, dx. \]


Incomplete.

One comment

  1. Anonymous says:

    For someone who may be seeking for solutions:

    (a) can be simply solved.
    (b) Some may use (a) to solve this question but this yields a wrong answer. Think of the definition of the denotation carefully.

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