0 < xⁿ/(1 + x) < xⁿ
0 < ∫(0→1) xⁿ/(1 + x) dx < ∫(0→1) xⁿ dx = xⁿ⁺¹/(n + 1) |(0→1) = 1/(n + 1)
∵lim(n→∞) 1/(n + 1) = 0
∴lim(n→∞) ∫(0→1) xⁿ/(1 + x) dx = 0
0 ≤ |∫(n→n + k) (sinx)/x dx| ≤ ∫(n→n + k) |sinx|/|x| dx ≤ ∫(n→n + k) 1/n dx = k/n
∵lim(n→∞) k/n = 0
∴lim(n→∞) ∫(n→n + k) (sinx)/x dx = 01/n应为1/x吧