Recall the definitions of the asymptotic notations. We will say that f(r) has "order of growth as r ro" (where o is either some fixed real number or too) if f(r) = e(r®) as r ro. 

1. Consider the functions f(r) = r sin r and g(r) = z. Is f(r) = e(g(r)) as r oo? Why or why not? (Hint: As always, you should refer back carefully to the definition of e(-).) 

2. Suppose that we know that f(r) = r +e(r) and g(r) = e(r) 0 as r 0. Determine the order of growth of f(r) +9(r). (This problem is meant to get you comfortable with manipulating asymptotic notation when it appears in expressions. When I say something like "f(r) = r+ O(r*", this means that there is some function h(r) = O(r), and f(r) = r+h(r). That is, the fact that h(r) = e(r) is the only thing you know about h(r).) 

3. Suppose that we know that f(r) = ea)as r o. Does this imply that f(r) = e()? (Hint: Think carefully about the definition of e), and consider f(r) = e.)