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

k∈ ..............

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- 1 year agoFavorite Answer
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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