The limit b→∞limbe−b arises in the evaluation of certain improper integrals. Which of the following best explains why this limit equals 0?
a) Because e−b→0 as b→∞, multiplying it by any quantity gives 0, so be−b→0.
b) Since the linear term b increases without bound, it eventually overrides the decay of e−b, causing be−b to diverge to ∞.
c) An exponential function decays faster than any polynomial grows, so e−b dominates b and their product approaches 0.
d) Applying L’Hôpital’s Rule once gives b→∞limebb=eb1→0, so the original limit must equal 1.