1. 应用 `epsilon`-`delta` 语言证明以下极限:
    1. `lim_(x to 2) x^2 = 4`;
    2. `lim_(x to 1) x/(2x^2 + 1) = 1/3`;
    3. `lim_(x to 1) (x^2 - 1)/(2x^2 - x - 1) = 2/3`;
    4. `lim_(x to 1) (x + 2)/(2 sqrt x - 1) = 3`;
    5. `lim_(x to pi/2) (2x-pi) cos((x-pi)/(2x-pi)) = 0`.
  2. 设 n 为正整数. 应用 `epsilon`-`delta` 语言证明以下极限:
    1. `lim_(x to x_0) x^n = x_0^n`;
    2. `lim_(x to x_0) x^(1/n) = x_0^(1/n)` (`x_0 gt 0`).
  3. 已知 `lim_(x to x_0) f(x) = a`. 应用 `epsilon`-`delta` 语言证明以下结论:
    1. `lim_(x to x_0) f^2(x) text(sgn) f(x) = a^2 text(sgn) a`;
    2. `lim_(x to x_0) root 3 f(x) = root 3 a`.
  4. 给出定理 3.1.4 结论 (2) 的证明.
  5. 证明函数极限 `lim_(x to x_0) f(x) = a` 的定义有以下各个等价的表述形式:
    1. `AA epsilon gt 0`, `EE n in NN`, 当 `0 lt |x - x_0| lt 1/n` 时, `|f(x) - a| lt epsilon`;
    2. `AA n in NN`, `EE delta gt 0`, 当 `0 lt |x - x_0| lt delta` 时, `|f(x) - a| lt 1/2^n`;
    3. `AA n in NN`, `EE m in NN`, 当 `0 lt |x - x_0| lt 1/m` 时, `|f(x) - a| lt 1/2^n`;
    4. `EE C gt 0`, `AA epsilon gt 0`, `EE delta gt 0`, 当 `0 lt |x - x_0| lt delta` 时, `|f(x) - a| lt C epsilon`.
  6. 应用函数极限的四则运算规律求以下极限:
    1. `lim_(x to 0) ((x-1)^3 - 2x - 1)/(x^3 + x - 2)`;
    2. `lim_(x to 1) (x^3 - 1)/(x^2 - 3x + 2)`;
    3. `lim_(x to 0) ((1+x)(1+2x)(1-3x) - 1)/(2x^3 + x^2)`;
    4. `lim_(x to -1) ((2+x)^4 - (5 + 4x))/((x+1)^2(x^2 + 2x + 3))`;
    5. `lim_(x to 1) (x^3 - 3x + 2)/(x^4 - 4x + 3)`;
    6. `lim_(x to -1) (x^3 - 3x - 2)/(x^5 - 2x - 1)`;
    7. `lim_(x to 4) (sqrt(1 + 2x) -3)/(sqrt x - 2)`;
    8. `lim_(x to -1) (2 - sqrt(x+5))/(1 + root 3 x)`;
    9. `lim_(x to 1)(sqrt(2-x) - sqrt x)/(root 3 (2-x) - root 3 x)`;
    10. `lim_(x to a) (sqrt(x-a) + sqrt x - sqrt a)/sqrt(x^2-a^2)`;

      11. (以下各题 m, n 均表示自然数)
    11. `lim_(x to 1) (x + x^2 + cdots + x^n - n)/(x - 1)`;
    12. `lim_(x to 1) (x^m - 1)/(x^n - 1)`;
    13. `lim_(x to 1) (x^(n+1) - (n+1)x + n)/(x-1)^2`;
    14. `lim_(x to 1) (m/(x^m - 1) - n/(x^n - 1))`.
  7. 已知 `lim_(x to x_0) f(x) = a`, `lim_(x to x_0) g(x) = b`. 证明:
    1. `lim_(x to x_0) max{f(x), g(x)} = max{a, b}`, `lim_(x to x_0) min{f(x), g(x)} = min{a, b}`;
    2. `lim_(x to x_0) arccos f(x) = arccos a` (设 `|f(x)| lt 1`, `|a| lt 1`), `lim_(x to x_0) "arccot" f(x) = "arccot" a`.
  8. 求以下极限:
    1. `lim_(x to x_0) e^(ax) "lg"(1+x^2)(cos bx + sin bx)`;
    2. `lim_(x to x_0) x/2 sqrt(a^2-x^2) + a^2/x arcsin {:x/a:}` (`|x_0| lt a`);
    3. `lim_(x to x_0) 2^(sqrt x) x^(3x^2)` (`x_0 gt 0`);
    4. `lim_(x to x_0) x^(a^x) + x^(x^a) + a^(x^x)` (`a gt 0, x_0 gt 0`).
  9. 设已知 `lim_(x to x_0) f(x) = a` 且 `lim_(x to a) g(x) = b`. 问是否能够据此推知 `lim_(x to x_0) g(f(x)) = b`? 考察例子: `f(x) = {x, 当 x 是有理数; 0, 当 x 是无理数:}`, `g(x) = {1, 当 x != 0; 0, 当 x = 0:}`. 问能否从 `lim_(x to 0) f(x) = 0` 和 `lim_(x to 0) g(x) = 1` 推出 `lim_(x to 0) g(f(x)) = 1`? 定理 3.1.7 的哪些条件不满足? 定理 3.1.8 的哪些条件不满足?
  10. 应用海涅定理, 从函数的极限求以下数列的极限:
    1. `lim_(n to oo) ( (4 cos^2 {:1/sqrt(n):}) / (cos^2 {:1/sqrt(n):} - 1) - (cos {:1/sqrt(n):} + 1) / (cos {:1/sqrt(n):} - 1) )`;
    2. `lim_(n to oo) (root m (1 + sin {:pi/n:}) - 1) / sin{:pi/n:}`;
    3. `lim_(n to oo) ln cos {:(pi(n^2-1))/(4(n^2+1)):}`;
    4. `lim_(n to oo) ln(1 + sqrt(n^2+1)) - ln n`.
  11. 给出定理 3.1.12 的证明.