7: You should get

Now it is time for a careful evaluation of terms. What do they do when x grows to infinity?

Remark: If you tried the alternative and applied the l'Hospital rule to the original fraction, you should have got

This does not really look that nice. Well, the cosine is nice, because when x goes to infinity, it becomes cos(0) = 1, so this can be figured out separately. What about the numerator? Since the root behaves like x, the numerator is approximately x² − x², which shows two things: It is the indeterminate difference ∞ − ∞ and factoring out wouldn't help as the two terms are of the same type. How to handle this? Try common denominator and some algebra to make it nicer.

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