7: It is better to "put up" the sine, since derivative of a root is worse than derivative of a sine. You then get

As expected, you obtained and indeterminate ratio and the l'Hospital rule is the next step.

Remark: If you decided to investigate the original numerator, you should have done

Now it is confirmed that the original ratio is of the type "zero over zero" and the l'Hospital rule can be used.

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