6: First of all, since hyperbolic sine is a "nice" function, it is possible to ignore it first and just figure out the limit of the ratio inside.

When n grows to infinity, the "−1" term can be ignored. So in the ratio you are in fact comparing the general power nⁿ with the logarithm, and we know from the scale of powers that the power eventually prevails. Thus your guess should be that the ratio converges to infinity.

To prove it, try to use the l'Hospital rule.

Then check what happens to hyperbolic sine at infinity.

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