In the following, remember that f(n)(x) represents the nth derivative of f, and assume s > 0.
(a) By giving an appropriately diverse selection of samples functions f, explain why it is reasonable to assume that for “most” functions f, there is some value s for which lim x→∞ f(x)e−sx = 0. (In other words, pretend you’re the teacher of a diﬀerential equations class trying to convince the class that this assumption is a reasonable one! Don’t just claim the limit is 0; show that the limit is 0 for your chosen example functions) (b) Suppose lim x→∞ f(x)e−sx = 0. Find lim x→∞ f(n)(x)e−sx. Suggestion: Visit the l’Hˆopital. (c) Suppose lim x→∞ f(n)(x)e−sx = 0. Find lim x→∞ f(x)e−sx. Suggestion: Your intuition should suggest it’s 0. What if it isn’t?