31: You should get

Now the infinity on the top is clear, but we cannot assign any limit to the denominator due to the oscillation of cosine. Actually, there is an even bigger problem. Note that the expression in the denominator is often zero, namely for x = 2πk. Such points go to infinity, so the given expression does not exist on any neighborhood of infinity and limit there makes no sense. We see that l'Hospital's rule gave a limit that does not exist, which means that this new limit actually need not have anything in common with the original limit (l'Hospital only works if it yields an existing limit). This is therefore a dead end.

The only reasonable hope lies in getting rid of that sine before we do l'Hospital's rule, that is, in combining comparison with l'Hospital. Find a suitable lower estimate for the given expression to force it up.

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