Decide which of the following statements are true and which are false. Prove the true ones and provide counterexamples for the false ones.
a) x/ log x → 0 as x → 0.
b) If n ∈ N, then sin(1/x)/xn → 0 as x →∞.
c) xlog x → 0 as x → 0+.
d) If there is a β > 0 such that f ′(x) ≥ β for all x ∈ (0,∞), then x2/ f (x)→∞as x →∞.
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