Recall the definitions of the asymptotic notations. We will say that f(x) has "order of growth x α as x → x0" (where x0 is either some fixed real number or ±[infinity]) if f(x) = Θ(x α) as x → x0. 1. Consider the functions f(x) = x sin x and g(x) = x. Is f(x) = Θ(g(x)) as x → [infinity]? Why or why not? (Hint: As always, you should refer back carefully to the definition of Θ(·).) 2. Suppose that we know that f(x) = x + Θ(x 2 ) and g(x) = Θ(x) > 0 as x → 0. Determine the order of growth of f(x) + g(x). (This problem is meant to get you comfortable with manipulating asymptotic notation when it appears in expressions. When I say something like "f(x) = x + Θ(x 2 )", this means that there is some function h(x) = Θ(x 2 ), and f(x) = x+h(x). That is, the fact that h(x) = Θ(x 2 ) is the only thing you know about h(x).) 3. Suppose that we know that f(x) = e Θ(x) as x → [infinity]. Does this imply that f(x) = Θ(e x )? (Hint: Think carefully about the definition of Θ(·), and consider f(x) = e 2x .)