1. 求以下函数的上、下极限:
    1. `x_n = (-1)^n root n n + 1/root n n`;
    2. `x_n = n/(n+1) sin((n pi)/2 + pi/3)`;
    3. `x_n = root n (1 + 2^(n (-1)^n))`;
    4. `x_n = (n^2)/(n^2+1) cos^2 ((n pi)/4)`;
    5. `x_n = (2(-1)^(sigma(n)) + 1/n) sin ((n pi)/2)`, `sigma(n)` 表示 n 的素因子个数.
    1. 设 `{x_n}` 是有界数列, m 是正整数. 证明:

      `underset(n to oo) text(lim inf) x_n^m = (underset(n to oo) text(lim inf) x_n)^m`,

      `underset(n to oo) text(lim sup) x_n^m = (underset(n to oo) text(lim sup) x_n)^m`;

    2. 设正数列 `{x_n}` 有界且有正的下界. 证明:

      `underset(n to oo) text(lim inf) 1/x_n = 1/(underset(n to oo) text(lim sup) x_n)`,

      `underset(n to oo) text(lim sup) 1/x_n = 1/(underset(n to oo) text(lim inf) x_n)`;

  3. 应用上、下极限求以下数列的极限:
    1. `x_1 = 0`, `x_(n+1) = 1/3 (x_n + 2)`, `n = 1, 2, cdots`;
    2. `x_1 = 0`, `x_(n+1) = 1/2 x_n^2 - 1`, `n = 1, 2, cdots`;
    3. `x_1 = sqrt 3`, `x_(n+1) = sqrt(3 x_n)`, `n = 1, 2, cdots`;
    4. `x_1 = sqrt 3`, `x_(n+1) = sqrt(3+x_n)`, `n = 1, 2, cdots`;
    5. `x_1 = 1`, `x_(n+1) = (3x_n - 1)/(x_n + 1)`, `n = 1, 2, cdots`;
    6. `x_1 = a in [0,1)`, `x_(n+1) = 1/2 (a + x_n^2)`, `n = 1,2,cdots`.
  4. 应用上、下极限求以下数列的极限:
    1. `x_1 = 2`, `x_(n+1) = (3 + 2x_n)/(2 + 3x_n)`, `n = 1, 2, cdots`;
    2. `x_1 = 3`, `x_(n+1) = 3 + 1/x_n`, `n = 1, 2, cdots`;
    3. `0 lt x_1 lt 1`, `x_(n+1) = x_n(2-x_n)`, `n = 1, 2, cdots`;
    4. `x_1 gt 0`, `x_(n+1) = (x_n(x_n^2 + 3a))/(3x_n^2 + a)`, `n = 1, 2, cdots` (a 为正常数);
    5. `x_1 gt 0`, `x_(n+1) = (m-1)/m x_n + a/m x_n^(1-m)`, `n = 1, 2, cdots` (a 为正常数, m ≥ 2 为正常数);
    6. `x_1 = a in [0,1]`, `x_(n+1) = 1/2 (a - x_n^2)`, `n = 1, 2, cdots`.
    1. 设 `0 lt x_1 lt y_1`, 且
      `x_(n+1) = sqrt(x_n y_n)`, `y_(n+1) = 1/2 (x_n + y_n)`, `n = 1, 2, cdots`.
      应用上、下极限证明: `{x_n}` 和 `{y_n}` 都有极限并且相等;
    2. 设 `x_1 = y_1 = 1`, 且
      `x_(n+1) = x_n + 2y_n`, `y_(n+1) = x_n + y_n`, `n = 1, 2, cdots`.
      应用上、下极限证明: `lim_(n to oo) y_n` 存在当且仅当 `lim_(n to oo) x_n` 存在.
  6. 应用上、下极限证明柯西收敛准则.
  7. 应用区间套定理和定理 2.5.3 证明定理 2.5.2.
    1. 设数列 `{x_n}` 有界并满足条件: `lim_(n to oo) (x_(n+1) + 2x_n) = 1`. 证明: `x_n` 收敛并求极限;
    2. 设数列 `{x_n}` 有界并满足条件: `lim_(n to oo) (x_(2n) + 2x_n) = 1`. 证明: `x_n` 收敛并求极限;
    3. 如果把上题中的 `x_(2n) + 2x_n` 换为 `x_n + 2x_(2n)`, 问结论是否成立? 举例说明;
    4. 设数列 `{x_n}` 满足以下两个条件:
      `x_n = O(1/n)`, `lim_(n to oo) n(x_(2n) + x_n) = c`, c 为常数.
      证明: 极限 `lim_(n to oo) n x_n` 存在, 并求其值.
  9. 设 `lim_(n to oo) (x_(n+1) - x_n) = a`. 证明: `lim_(n to oo) x_n / n = a`.
  10. 设正数列 `{x_n}` 收敛且极限为 a. 令 `y_n = root n(x_1 x_2 cdots x_n)`, `n = 1, 2, cdots`. 证明: 数列 `{y_n}` 也收敛且极限是 a.