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Math Help - limit + integral

  1. #1
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    limit + integral

    Hey,
    Any help would be appreciated.
    I'm not even sure where to start.

    Compute


    lim (x^2)/x-3 [Integral] (from 3 to x) (sin t)/t dt
    x->3
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  2. #2
    MHF Contributor
    skeeter's Avatar
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    L'Hopital?

    \lim_{x \to 3} \frac{x^2 \cdot \frac{\sin{x}}{x} + 2x \cdot \int_3^x \frac{\sin{t}}{t}\,dt}{1}
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  3. #3
    Math Engineering Student
    Krizalid's Avatar
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    Power series allowed?

    I haven't checked it well though.
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  4. #4
    o_O
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    Let F(x) = \int_{3}^{x} \frac{\sin t}{t} \ dt.

    Note: F(3) = 0

    So: \lim_{x \to 3} \frac{x^2}{x-3} \cdot \int_{3}^{x} \frac{\sin t}{t} \ dt \ = \ \lim_{x \to 3} x^2 \cdot \frac{F(x) - F(3)}{x - 3} \ = \ \lim_{x \to 3} x^2 \ \cdot \  \underbrace{\lim_{x \to 3} \frac{F(x) - F(3)}{x - 3}}_{F'(3)} \ = \  \hdots
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