16: 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.
Next hint
Answer