For what values of k is limx→∞(coshkx)/(sinh3x) finite?

k∈ ..............

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  • 10 months ago
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    sinh(kx) / cosh(3x) = [ e^(kx) - e^(-kx) ] / [ e^(3x) + e^(-3x) ]

    When x is large and assume k >= 0

    sinh(kx) / cosh(3x) ≈ e^(kx) / e^(3x) = e^(x(k - 3))

    For sinh(kx) / cosh(3x) to be finite

    e^(x(k - 3)) is finite

    ==> k - 3 <= 0

    ==> k <= 3

    ==> 0 <= k <= 3

    Similarly, when x is large, and k <= 0

    sinh(kx) / cosh(3x) ≈ -e^(-kx) / e^(3x) = -e^(x(-k - 3))

    -e^(x(-k - 3)) is finite

    ==> -k - 3 <= 0

    ==> k + 3 >= 0

    ==> k >= -3

    Combining both scenarios gives the range of k of [-3, 3]

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