1. `lim_(x to 0^+) (x-1)/(3x-2sqrt x - 1)`;
2. `lim_(x to 1^-) (root 3 (1-x) - root 4 (1-x)) / (root 3 (1-x) + 3 root 4 (1-x))`;
3. `lim_(x to 1^+) ([3x])/(x+2)`;
4. `lim_(x to 1^-) ([3x])/(x+2)`;
5. `lim_(x to 2^+) ([x]^2 - 1)/(x^2-1)`;
6. `lim_(x to 2^-) ([x]^2 - 1)/(x^2-1)`;
7. `lim_(x to 2^+) arctan {:sqrt(x-1)/(x-2):}`;
8. `lim_(x to 2^-) arctan {:sqrt(x-1)/(x-2):}`;
9. `lim_(x to 0^+) 1/(1+2^(1/x))`;
10. `lim_(x to 0^-) 1/(1+2^(1/x))`.

1. `1`;
2. `-1/3`;
3. `1`;
4. `2/3`;
5. `1`;
6. `0`;
7. `pi/2`;
8. `-pi/2`;
9. `0`;
10. `1`.

第 (2) 题可令 `y = root 12 (1-x)`, 转化为 `y to 0^+` 的极限.

1. `lim_(x to +oo) (sqrt((x+a)(x+b)) - x)`;
2. `lim_(x to +oo) (root 3 (x^3+3x^2) - sqrt(x^2-2x))`;
3. `lim_(x to +oo) (sqrt(x+sqrt(x+sqrt x)) - sqrt(x-sqrt(x+sqrt x)))`;
4. `lim_(x to oo) ((x+sqrt(x^2-2x))^n + (x-sqrt(x^2-2x))^n) / (root 3 (x^(3n)+1) + root 3 (x^(3n)-1))`;
5. `lim_(x to oo) x^(1/3) [(x+1)^(2/3) - (x-1)^(2/3)]`;
6. `lim_(x to oo) arcsin {:(1-x)/(1+x):}`;
7. `lim_(x to +oo) (sin sqrt(x+1) - sin sqrt x)`;
8. `lim_(x to +oo) arccos(sqrt(x^2+x)-x)`.

1. 原式 = `lim_(x to +oo) ((x+a)(x+b)-x^2)/(sqrt((x+a)(x+b))+x)` `= lim_(x to +oo) (a+b+(ab)/x)/(sqrt((1+a/x)(1+b/x))+1)` `= (a+b)/2`;
2. 原式 = `lim_(x to +oo) ((x^3+3x^2)^2 - (x^2-2x)^3) / (sum_(k=0)^5 (x^2+3x^2)^(k/3) (x^2-2x)^((5-k)/2))` `= (12x^5 + o(x^5))/(6x^5 + o(x^5))` `= 2`;
3. 原式 = `lim_(x to +oo) (2sqrt(x+sqrt x))/(sqrt(x+sqrt(x+sqrt x)) + sqrt(x-sqrt(x+sqrt x)))` `= lim_(x to +oo) (2(sqrt x + o(sqrt x)))/(2(sqrt x + o(sqrt x)))` `= 1`;
4. 原式 = `lim_(x to oo) ((2x + o(x))^n + o(x^n))/(2x^n + o(x^n))` `= 2^(n-1)`;
5. 原式 = `lim_(x to oo) (root 3 (x^3+2x^2+x) - root 3 (x^3-2x^2+x))` `= lim_(x to oo) (4x^2)/(3x^2 + o(x^2))` `= 4/3`;
6. 原式 = `arcsin(-1) = -pi/2`;
7. 因为 `sqrt(x+1) - sqrt x = 1/(sqrt(x+1) + sqrt x) to 0`, 我们有原式 = `lim_(x to +oo) 2 sin {:(sqrt(x+1) - sqrt x)/2:} cos {:(sqrt(x+1) + sqrt x)/2:} = 0`;
8. 因为 `sqrt(x^2+x) - x = x/(sqrt(x^2+x)+x) to 1/2`, 我们有原式 = `arccos 1/2 = pi/3`.

1. `lim_(x to 0) (sin ax)/(sin bx)` (`a, b != 0`);
2. `lim_(x to 0) (tan ax)/(tan bx)` (`a, b != 0`);
3. `lim_(x to 0) (sin 3x - sin 2x)/(sin 5x)`;
4. `lim_(x to 0) (1-cos x)/x^2`;
5. `lim_(x to 0) (tan x - sin x)/(sin^3 x)`;
6. `lim_(x to 0) (cos 3x - cos 2x)/(sin x^2)`;
7. `lim_(x to pi/4) tan 2x tan (pi/4 - x)`;
8. `lim_(x to 1) (1-x) tan {:(pi x)/2:}`;
9. `lim_(x to 0) sqrt(1- cos x^2)/(1-cos x)`;
10. `lim_(x to 0^+) (1-sqrt(cos x))/(1-cos sqrt x)^2`;
11. `lim_(x to 0) (arctan x)/(arcsin x)`;
12. `lim_(n to oo) cos{:x/2:} cos{:x/4:} cdots cos{:x/2^n:}`.

1. 原式 = `lim_(x to 0) (ax)/(bx) = a/b`;
2. 原式 = `lim_(x to 0) (ax)/(bx) = a/b`;
3. 原式 = `lim_(x to 0) (2sin{:x/2:}cos{:(5x)/2:})/(sin 5x) = 1/5`;
4. 原式 = `lim_(x to 0) (1-cos2x)/(4x^2) = lim_(x to 0) (2 sin^2 x)/(4x^2) = 1/2`;
5. 原式 = `lim_(x to 0) (tan x(1-cos x))/x^3 = 1/2`;
6. 原式 = `lim_(x to 0) (-2sin{:(5x)/2:}sin{:x/2:})/x^2 = -5/2`;
7. 原式 = `lim_(x to pi/4) (2tan x)/(1-tan^2 x) (1-tan x)/(1+tan x) = lim_(x to pi/4) (2tan x)/(1+tan x)^2 = 1/2`;
8. 原式 = `lim_(x to 0) x tan{:(pi(1-x))/2:} = lim_(x to 0) x/tan{:(pi x)/2:} = 2/pi`;
9. 原式 = `lim_(x to 0) sqrt(1-cos(4x^2))/(1-cos 2x) = lim_(x to 0) (sqrt 2 sin(2x^2))/(2 sin^2 x) = sqrt 2`;
10. 原式 = `lim_(x to 0^+) (1-cos x)/((1-cos sqrt x)^2(1+sqrt cos x)) = lim_(x to 0^+) (1/2 x^2)/((1/2 x)^2 * 2) = 1`;
11. 原式 = `lim_(x to 0) x/x = 1`;
12. `cos{:x/2:} cos{:x/4:} cdots cos{:x/2^n:} sin{:x/2^n:} = sin x/2^n`, 故原式 = `lim_(n to oo) sin x/(2^n sin{:x/2^n:}) = sin x / x`.

1. `lim_(x to 0) (1-2x)^(1/x)`;
2. `lim_(x to 0) (e^x + x)^(1/x)`;
3. `lim_(x to oo) ((x+a)/(x-a))^x`;
4. `lim_(x to oo) ((x^2+1)/(x^2-1))^x`;
5. `lim_(x to 0) ln(x^2+e^x) / ln(x^3+e^(2x)`;
6. `lim_(x to 0) ln(1+xe^x) / ln(x+sqrt(1+x^2))`;
7. `lim_(x to 0) (2e^x-1)^(1/x)`;
8. `lim_(x to 0) (2e^(x/(1+x))-1)^((1+x^2)/x)`;
9. `lim_(x to -oo) ln(1+3^x) / ln(1+2^x)`;
10. `lim_(x to +oo) ln(1+3^x) / ln(1+2^x)`;
11. `lim_(x to +oo) ln(1+2^x) ln(1+3/x)`;
12. `lim_(x to 1) (1-x) log_x 2`;

1. 原式 = `lim_(x to 0) e^(-2x * 1/x) = e^-2`;
2. 原式 = `lim_(x to 0) e(1+x/e^x)^(1/x) = e^2`;
3. 原式 = `lim_(x to oo) (1+(2a)/(x-a))^x = e^(2a)`;
4. 原式 = `lim_(x to oo) (1+2/(x^2-1))^x = 1`;
5. 原式 = `lim_(x to 0) (x+ln(1+x^2 e^-x))/(2x+ln(1+x^3 e^(-2x))) = lim_(x to 0) (x+o(x))/(2x+o(x)) = 1/2`;
6. 原式 = `lim_(x to 0) ln(1+xe^x)/(1/2 ln(1+x^2) + ln(1+x/sqrt(1+x^2)))` `= lim_(x to 0) (xe^x + o(x))/(x/sqrt(1+x^2) + o(x))` `= 1`;
7. 原式 = `lim_(x to 0) (e^x + e^x - 1)^(1/x) = lim_(x to 0) e (1+(e^x-1)/e^x)^(1/x) = e^2`;
8. 原式 = `lim_(x to 0) (2e^(x/(1+x)) - 1)^{:(1+x)/x (1+x^2)/(1+x)`, 由第 (7) 小题, 结果为 `e^2`;
9. 原式 = `lim_(x to -oo) 3^x/2^x = 0`;
10. 原式 = `lim_(x to +oo) (x ln 3 + ln(1+3^-x))/(x ln 2 + ln(1+2^-x)) = ln3/ln2`;
11. 原式 = `lim_(x to +oo) (xln2 + ln(1+2^-x))ln(1+3/x)` `= lim_(x to +oo) xln2ln(1+3/x)` `= 3ln2`;
12. 原式 = `lim_(x to 0) (xln2)/ln(1-x) = -ln2`.

1. `lim_(x to 0) (1+x^2)^(cot^2 x)`;
2. `lim_(x to 1) (1+sin pi x)^(cot pi x)`;
3. `lim_(x to 0) (cos x / cos (2x))^(1/x^2)`;
4. `lim_(x to 0^+) (cos sqrt x)^(1/x)`;
5. `lim_(x to pi/4) (tan x)^(tan 2x)`;
6. `lim_(x to pi/2) (sin x)^(tan x)`;
7. `lim_(x to 0) (ln tan(pi/4 + ax)) / (sin bx)`;
8. `lim_(x to 0) (ln cos ax)/(ln cos bx)`;
9. `lim_(n to oo) cos^n {:x/sqrt n:}`;
10. `lim_(n to oo) tan^n (pi/4 + 1/n)`;
11. `lim_(n to oo) n(root n x - 1)` (`x gt 0`);
12. `lim_(n to oo) n^2 (root n x - root (n+1) x)` (`x gt 0`).

1. 原式 = `lim_(x to 0) (1+x^2)^(1/tan^2 x) = e`;
2. 原式 = `lim_(x to 0) (1+sin(pi(1-x)))^cot(pi(1-x))` `= lim_(x to 0) (1+sin pi x)^(-cos{:pi x:}/sin{:pi x:})` `= e^-1`;
3. 原式 = `lim_(x to 0) (cos x/(2cos^2 x - 1))^(1/x^2)` `= lim_(x to 0) (1+(1+cos x - 2cos^2 x)/(2cos^x - 1))^(1/x^2)` `= lim_(x to 0) exp(((1+2cos x)(1-cos x))/x^2)` `= e^(3/2)`;
4. 原式 = `lim_(x to 0^+) (1+cos x-1)^(1/x^2) = lim_(x to 0^+) e^((cos x-1)/x^2) = e^(-1/2)`;
5. 原式 = `lim_(x to pi/4) (1+tan x-1)^((2tan x)/(1-tan^2 x)) = lim_(x to pi/4) e^(-(2tan x)/(1+tan x)) = e^-1`;
6. 原式 = `lim_(x to 0) (1+cos x-1)^cot x = lim_(x to 0) e^((cos x-1)/tan x) = 1`;
7. 原式 = `lim_(x to 0) ln{:(1+tan ax)/(1-tan ax):}/sin{:bx:} = lim_(x to 0) ln(1+(2tan ax)/(1-tan ax))/sin{:bx:} = (2a)/b`;
8. 原式 = `lim_(x to 0) ln(1+cos ax-1)/ln(1+cos bx-1) = a^2/b^2`;
9. 原式 = `lim_(n to oo) (1+cos{:x/n:}-1)^(n^2) = lim_(n to oo) e^(n^2(cos{:x/n:}-1)) = e^(-x^2/2)`;
10. 原式 = `lim_(n to oo) ((1+tan{:1/n:})/(1-tan{:1/n:}))^n` `= lim_(n to oo) (1+(2tan{:1/n:})/(1-tan{:1/n:}))^n` `= lim_(n to oo) e^((2n tan{:1/n:})/(1-tan{:1/n:})) = e^2`;
11. 原式 = `lim_(n to oo) n(e^(ln x/n) -1) = ln x`;
12. 原式 = `lim_(n to oo) n^2(e^(ln x/n) - e^(ln x/(n+1))) = lim_(n to oo) n^2 e^(ln x/(n+1))(e^(ln x/(n(n+1))) - 1) = ln x`.

1. `lim_(x to 0) (a^x-1)/x` (`a != 1`);
2. `lim_(x to 0) ((a^x + b^x)/2)^(1/x)`;
3. `lim_(x to 0) ((a^(x^2) + b^(x^2))/(a^x+b^x))^(1/x)`;
4. `lim_(x to 0) ((a^x + b^x + c^x)/(a+b+c))^(1/x)`;
5. `lim_(n to oo) n(root n a - 1)`;
6. `lim_(n to oo) n^2(root n a - root (n+1) a)`;
7. `lim_(n to oo) ((root n a + root n b + root n c)/ 3)^n`;
8. `lim_(n to oo) ((a - 1 + root n b)/a)^n`.

1. 原式 = `lim_(x to 0) (e^(x ln a)-1)/x = ln a`;
2. 原式 = `lim_(x to 0) (1+(a^x+b^x-2)/2)^(1/x)` `= lim_(x to 0) exp{:(a^x-1+b^x-1)/(2x):}` `= lim_(x to 0) exp{:(x ln (ab) + o(x))/(2x):}` `= sqrt(ab)`;
3. 原式 = `lim_(x to 0) (1+(a^(x^2) + b^(x^2) -a^x-b^x)/(a^x+b^x))^(1/x)` `= lim_(x to 0) exp{:((x^2-x)ln(ab)+o(x))/(2x):}` `= 1/sqrt(ab)`;
4. `a+b+c != 3` 时, 极限不存在;
5. 由 6(11), 原式 = `ln a`.
6. 由 6(12), 原式 = `ln a`.
7. 原式 = `lim_(n to oo) (1+(root n a + root n b + root n c - 3)/3)^n` `= lim_(n to oo) exp((ln(abc)/n + o(1/n))/3 * n)` `= root 3 (abc)`;
8. 原式 = `lim_(n to oo) (1+(root n b - 1)/a)^n = lim_(n to oo) exp(ln b/(na) * n) = b^(1/a)`.

1. `arcsin x ~ x`;
2. `arctan x ~ x`;
3. `e^x - 1 ~ x`;
4. `1 - cos x ~ 1/2 x^2`.

1. 令 `x = sin y`;
2. 令 `x = tan y`;
3. 令 `x = ln(1+y)`;
4. 见 4(4);

1. `lim_(x to 0) (sqrt(1+x^2) - 1)/(1-cos x)`;
2. `lim_(x to 0) (x ln(1+x^2))/(sin x^2 tan x)`;
3. `lim_(x to 0) (x e^(2x) sin x)/(e^x - e^-x)^2`;
4. `lim_(x to oo) (x^2 arctan {:1/x:})/(2x - cos x)`.

1. `1`;
2. `1`;
3. `1/4`;
4. `1/2`.

1. 推导曲线在正、负无穷远处存在渐近线的充分必要条件;
2. 求下列函数在正、负无穷远处的渐近线:
   1. `y = sqrt(x^2 - x + 1)`,
   2. `y = (x^2+1)/(x+1)`,
   3. `y = ln(1+e^x)`,
   4. `y = x + arccos {:1/(x^2+1):}`,
   5. `y = (xe^x)/(e^x+1)`,
   6. `y = x^(x+1)/(1+x)^x`.

2. 1. `+oo`: `y = x - 1/2`, `-oo`: `y = -x + 1/2`;
   2. `oo`: `y = x-1`;
   3. `+oo`: `y = x`, `-oo`: `y = 0`;
   4. `oo`: `y = x + pi/2`;
   5. `+oo`: `y = x`, `-oo`: `y = 0`;
   6. `oo`: `y = x/e + 1/(2e)`;

1. 如果存在数列 `{x_n}` 使 `lim_(n to oo) x_n = +oo` 且 `lim_(n to oo) f(x_n) = a`, 则 `lim_(x to +oo) f(x) = a`;
2. 如果 `f` 严格单增, `lim_(x to +oo) f(x) = a` 且 `lim_(n to oo) f(x_n) = a`, 则 `lim_(n to oo) x_n = +oo`.

1. 先证 `f` 有界. 如若不然, 设对任意 `M gt 0`, `EE x_0 in (b, +oo)` `f(x_0) gt M`. 由 `x_n to +oo` 知 `EE n_0 in NN`, `x_(n_0) gt x_0`, 从而 `f(x_(n_0)) ge f(x_0) gt M`, 与 `f(x_n) to a` 矛盾. 现在由单调有界原理知 `lim_(x to +oo) f(x)` 存在, 再由 Heine 定理知极限等于 a.

1. `lim_(x to +oo) (f(x))/x = lim_(x to +oo) [f(x+1) - f(x)]`;
2. `lim_(x to +oo) [f(x)]^(1/x) = lim_(x to +oo) (f(x+1))/(f(x))`.