令t(x) = 1/x则
lim(x→∞)xsin1/x = lim (x→∞) 1/t(x) * sin t(x)
由于当x→∞时t(x)→0,因此
lim (x→∞) 1/t(x) * sin t(x) = lim(t→0) (sin t)/t = 1.