Задачник Кузнецова Пределы Задачи 1-4

Задача 1

Доказать, что $\lim_{n\to\infty} {a_{n}} = a$ (указать $N(\varepsilon)$).

Задача	Условие	Задача	Условие
1-1	$a_{n} = \frac {3n - 2} {2n - 1}, \ a = \frac {3} {2}$	1-2	$a_{n} = \frac {4n - 1} {2n + 1},\ a = 2$
1-3	$a_{n} = \frac {7n + 4} {2n + 1},\ a = \frac {7} {2}$	1-4	$a_{n} = \frac {2n - 5} {3n + 1},\ a = \frac {2} {3}$
1-5	$a_{n} = \frac {7n - 1} {n + 1},\ a = 7$	1-6	$a_{n} = \frac {4n^2 + 1} {3n^2 + 2},\ a = \frac {4} {3}$
1-7	$a_{n} = \frac {9 - n^3} {1 + 2n^3},\ a = - \frac {1} {2}$	1-8	$a_{n} = \frac {4n - 3} {2n + 1},\ a = 2$
1-9	$a_{n} = \frac {1 - 2n^2} {2 + 4n^2},\ a = - \frac {1} {2}$	1-10	$a_{n} = - \frac {5n} {n + 1},\ a = -5$
1-11	$a_{n} = \frac {n + 1} {1 - 2n},\ a = - \frac {1} {2}$	1-12	$a_{n} = \frac {2n + 1} {3n - 5},\ a = \frac {2} {3}$
1-13	$a_{n} = \frac {1 - 2n^2} {n^2 + 3},\ a = -2$	1-14	$a_{n} = \frac {3n^2} {2 - n^2},\ a = -3$
1-15	$a_{n} = \frac {n} {3n - 1},\ a = \frac {1} {3}$	1-16	$a_{n} = \frac {3n^3} {n^3 - 1},\ a = 3$
1-17	$a_{n} = \frac {4 + 2n} {1 -3n},\ a = - \frac {2} {3}$	1-18	$a_{n} = \frac {5n + 15} {6 - n},\ a = -5$
1-19	$a_{n} = \frac {3 - n^2} {4 + 2n^2},\ a = - \frac {1} {2}$	1-20	$a_{n} = \frac {2n - 1} {2 - 3n},\ a = - \frac {2} {3}$
1-21	$a_{n} = \frac {3n - 1} {5n + 1},\ a = \frac {3} {5}$	1-22	$a_{n} = \frac {4n - 3} {2n + 1},\ a = 2$
1-23	$a_{n} = \frac {1 - 2n^2} {2 + 4n^2},\ a = - \frac {1} {2}$	1-24	$a_{n} = \frac {5n + 1} {10n - 3},\ a = \frac {1} {2}$
1-25	$a_{n} = \frac {2 - 2n} {3 + 4n},\ a = - \frac {1} {2}$	1-26	$a_{n} = \frac {23 - 4n} {2 - n},\ a = 4$
1-27	$a_{n} = \frac {1 + 3n} {6 - n},\ a = -3$	1-28	$a_{n} = \frac {2n + 3} {n + 5},\ a = 2$
1-29	$a_{n} = \frac {3n^2 + 2} {4n^2 - 1},\ a = \frac {3} {4}$	1-30	$a_{n} = \frac {2 - 3n^2} {4 + 5n^2},\ a = - \frac {3} {5}$
1-31	$a_{n} = \frac {2n^3} {n^3 - 2},\ a = 2$

Задача 2

Вычислить предел числовой последовательности:

Задача	Условие	Задача	Условие
2-1	$\lim_{n\to\infty} \frac {(3 - n)^2 + (3 + n)^2} {(3 - n)^2 - (3 + n)^2}$	2-2	$\lim_{n\to\infty} \frac {(3 - n)^4 - (2 - n)^4} {(1 - n)^4 - (1 + n)^4}$
2-3	$\lim_{n\to\infty} \frac {(3 - n)^4 - (2 - n)^4} {(1 - n)^3 - (1 + n)^3}$	2-4	$\lim_{n\to\infty} \frac {(1 - n)^4 - (1 + n)^4} {(1 + n)^3 - (1 - n)^3}$
2-5	$\lim_{n\to\infty} \frac {(6 - n)^2 - (6 + n)^2} {(6 + n)^2 - (1 - n)^2}$	2-6	$\lim_{n\to\infty} \frac {(n + 1)^3 - (n + 1)^2} {(n - 1)^3 - (n + 1)^3}$
2-7	$\lim_{n\to\infty} \frac {(1 + 2n)^3 - 8n^3} {(1 + 2n)^2 + 4n^2}$	2-8	$\lim_{n\to\infty} \frac {(3 - 4n)^2} {(n - 3)^3 - (n + 3)^3}$
2-8 (2 вариант)	$\lim_{n\to\infty} \frac {(3 - 4n)^2} {(n - 3)^2 - (n + 3)^2}$	2-9	$\lim_{n\to\infty} \frac {(3 - n)^3} {(n + 1)^2 - (n + 1)^3}$
2-10	$\lim_{n\to\infty} \frac {(n + 1)^2 + (n - 1)^2 - (n + 2)^3} {(4 - n)^3}$	2-11	$\lim_{n\to\infty} \frac {2(n + 1)^3 - (n - 2)^3} {n^2 + 2n -3}$
2-12	$\lim_{n\to\infty} \frac {(n + 1)^3 + (n + 2)^3} {(n + 4)^3 + (n + 5)^3}$	2-13	$\lim_{n\to\infty} \frac {(n + 3)^3 + (n + 4)^3} {(n + 3)^4 - (n + 4)^4}$
2-14	$\lim_{n\to\infty} \frac {(n + 1)^4 - (n - 1)^4} {(n + 1)^3 + (n - 1)^3}$	2-15	$\lim_{n\to\infty} \frac {8n^3 - 2n} {(n + 1)^4 - (n - 1)^4}$
2-16	$\lim_{n\to\infty} \frac {(n + 6)^3 - (n + 1)^3} {(2n + 3)^2 + (n + 4)^2}$	2-17	$\lim_{n\to\infty} \frac {(2n - 3)^3 - (n + 5)^3} {(3n - 1)^3 + (2n + 3)^3}$
2-18	$\lim_{n\to\infty} \frac {(n + 10)^2 + (3n + 1)^2} {(n + 6)^3 - (n + 1)^3}$	2-19	$\lim_{n\to\infty} \frac {(2n + 1)^3 + (3n + 2)^3} {(2n + 3)^3 - (n - 7)^3}$
2-20	$\lim_{n\to\infty} \frac {(n + 7)^3 - (n + 2)^3} {(3n + 2)^2 + (4n + 1)^2}$	2-21	$\lim_{n\to\infty} \frac {(2n + 1)^3 - (2n + 3)^3} {(2n + 1)^2 + (2n + 3)^2}$
2-22	$\lim_{n\to\infty} \frac {n^3 - (n - 1)^3} {(n + 1)^4 - n^4}$	2-23	$\lim_{n\to\infty} \frac {(n + 2)^4 - (n - 2)^4} {(n + 5)^2 + (n - 5)^2}$
2-24	$\lim_{n\to\infty} \frac {(n + 1)^4 - (n - 1)^4} {(n + 1)^3 + (n - 1)^3}$	2-25	$\lim_{n\to\infty} \frac {(n + 1)^3 - (n - 1)^3} {(n + 1)^2 - (n - 1)^2}$
2-26	$\lim_{n\to\infty} \frac {(n + 1)^3 - (n - 1)^3} {(n + 1)^2 + (n - 1)^2}$	2-27	$\lim_{n\to\infty} \frac {(n + 2)^3 + (n - 2)^3} {n^4 + 2n^2 - 1}$
2-28	$\lim_{n\to\infty} \frac {(n + 1)^3 + (n - 1)^3} {n^3 - 3n}$	2-29	$\lim_{n\to\infty} \frac {(n + 1)^3 + (n - 1)^3} {n^3 + 1}$
2-30	$\lim_{n\to\infty} \frac {(n + 2)^2 - (n - 2)^2} {(n + 3)^2}$	2-31	$\lim_{n\to\infty} \frac {(2n + 1)^2 - (n + 1)^2} {n^2 + n + 1}$

Задача 3

Вычислить предел числовой последовательности:

Задача	Условие	Задача	Условие
3-1	$\lim_{n\to\infty} \frac {n\sqrt[3]{5n^2} + \sqrt[4]{9n^8 + 1}} {\left(n + \sqrt{n}\right)\sqrt{7 - n + n^2}}$	3-2	$\lim_{n\to\infty} \frac {\sqrt{n - 1} - \sqrt{n^2 + 1}} {\sqrt[3]{3n^3 + 3} + \sqrt[4]{n^5 + 1}}$
3-3	$\lim_{n\to\infty} \frac {\sqrt{n^3 + 1} - \sqrt{n - 1}} {\sqrt[3]{n^3 + 1} - \sqrt{n - 1}}$	3-4	$\lim_{n \to \infty } \frac{\sqrt[3]{n^2-1}+7n^3}{\sqrt[4]{n^{12}+n+1}-n}$
3-5	$\lim_{n \to \infty } \frac{\sqrt{3n-1} -\sqrt[3]{125n^3+n}}{\sqrt[5]{n}-n}$	3-5(2 вариант)	$\lim_{n \to \infty } \frac{\sqrt{3n-1} -\sqrt[3]{125n^3+n}}{\sqrt[3]{n}-n}$
3-6	$\lim_{n\to \infty } \frac{n \sqrt[5]{n}-\sqrt[3]{27n^6+n^2}}{\left( {n+\sqrt[4]{n}} \right)\sqrt{9+n^2}}$	3-7	$\lim_{n\to \infty } \frac{\sqrt{n+2} -\sqrt {n^2+2}}{\sqrt[4]{4n^4+1}-\sqrt[3]{n^4-1}}$
3-8	$\lim_{n\to \infty } \frac{\sqrt {n^4+2} +\sqrt{n-2} }{\sqrt[4]{n^4+2}+\sqrt {n-2}}$	3-9	$\lim_{n\to \infty} \frac{6n^3-\sqrt{n^5+1}} {\sqrt {4n^6+3} -n}$
3-10	$\lim_{n\to \infty } \frac{\sqrt{5n+2} -\sqrt[3]{8n^3+5}}{\sqrt[4]{n+7}-n}$	3-11	$\lim_{n\to \infty } \frac{n \sqrt[4]{3n+1}+\sqrt {81n^4-n^2+1} }{\left( {n+\sqrt[3]{n}} \right)\sqrt{5-n+n^2}}$
3-12	$\lim_{n\to \infty} \frac {\sqrt {n + 3} - \sqrt {n^2 -3}} {\sqrt[3]{n^5 - 4} - \sqrt[4]{n^4 + 1}}$	3-13	$\lim_{n\to \infty} \frac {\sqrt {n^5 + 3} - \sqrt {n - 3}} {\sqrt[5]{n^5 + 3} + \sqrt{n - 3}}$
3-14	$\lim_{n\to \infty} \frac{\sqrt[3]{n}-9n^2} {3n-\sqrt[4]{9n^8+1}}$	3-15	$\lim_{n\to \infty} \frac{\sqrt{4n+1} -\sqrt[3]{27n^3+4}}{\sqrt[4]{n}-\sqrt[3]{n^5+n}}$
3-16	$\lim_{n\to \infty} \frac{n\sqrt[3]{7n}-\sqrt[4]{81n^8-1}}{\left( {n+4\sqrt{n}}\right)\sqrt{n^2-5}}$	3-17	$\lim_{n\to \infty} \frac{\sqrt[3]{n^3-7}+\sqrt[3]{n^2+4}}{\sqrt[4]{n^5+5}+\sqrt{n}}$
3-18	$\lim_{n\to \infty} \frac{\sqrt{n^6+4} +\sqrt{n-4}}{\sqrt[5]{n^6+6}-\sqrt {n-6}}$	3-19	$\lim_{n\to \infty} \frac {4n^2-\sqrt[4]{n^3}} {\sqrt[3]{n^6+n^3+1}-5n}$
3-20	$\lim_{n\to \infty} \frac {\sqrt{n + 3} - \sqrt[3]{8n^3 + 3}} {\sqrt[4]{n + 4} - \sqrt[5]{n^5 + 5}}$	3-21	$\lim_{n\to \infty} \frac{n \sqrt[4]{11n}+\sqrt{25n^4-81}} {\left(n-7\sqrt {n}\right) \sqrt{n^2-n+1}}$
3-22	$\lim_{n\to \infty } \frac{\sqrt[3]{n^2}-\sqrt {n^2+5}} {\sqrt[5]{n^7}-\sqrt {n+1}}$	3-23	$\lim_{n\to \infty} \frac{\sqrt{n^7+5} -\sqrt{n-5}} {\sqrt[7]{n^7+5} + \sqrt{n-5}}$
3-24	$\lim_{n\to \infty} \frac{\sqrt[3]{n^2+2}-5n^2} {n-\sqrt{n^4-n+1}}$	3-25	$\lim_{n\to \infty} \frac{\sqrt{n+2} -\sqrt[3]{n^3+2}}{\sqrt[7]{n+2}-\sqrt[5]{n^5+2}}$
3-26	$\lim_{n\to \infty} \frac{n\sqrt{71n}-\sqrt[3]{64n^6+9}}{\left( {n-\sqrt[3]{n}} \right)\sqrt{11+n^2} }$	3-27	$\lim_{n\to \infty} \frac{\sqrt{n+6} -\sqrt {n^2-5} }{\sqrt[3]{n^3+3}+\sqrt[4]{n^3+1}}$
3-28	$\lim_{n\to \infty} \frac {\sqrt{n^8 + 6} - \sqrt{n-6}} {\sqrt[8]{n^8 + 6} + \sqrt{n - 6}}$	3-29	$\lim_{n\to \infty} \frac{n^2-\sqrt{n^3+1}} {\sqrt[3]{n^6+2}-n}$
3-30	$\lim_{n\to \infty} \frac{\sqrt{n+1} - \sqrt[3]{n^3+1}}{\sqrt[4]{n+1}-\sqrt[5]{n^5+1}}$	3-31	$\lim_{n\to \infty} \frac{n \sqrt[6]{n} + \sqrt[3]{n^{10}+1}} {\left( {n+\sqrt[4]{n}} \right)\sqrt[3]{n^3-1}}$
3-31 (2 вариант)	$\lim_{n\to \infty} \frac{n \sqrt[6]{n} + \sqrt[3]{32n^{10}+1}} {\left( {n+\sqrt[4]{n}} \right)\sqrt[3]{n^3-1}}$

Задача 4

Вычислить предел числовой последовательности:

Задача	Условие	Задача	Условие
4-1	$\lim_{n\to \infty} n\left( {\sqrt{n^2+1} +\sqrt {n^2-1} } \right)$	4-2	$\lim_{n\to \infty} n\left( {\sqrt {n\left( {n-2} \right)} -\sqrt {n^2-3}} \right)$
4-3	$\lim_{n\to \infty} \left(n-\sqrt[3]{n^3-5}\right)n\sqrt{n}$	4-4	$\lim_{n\to \infty} \left({\sqrt{\left(n^2+1\right)\left(n^2-4\right)}-\sqrt{n^4-9}}\right)$
4-5	$\lim_{n\to \infty} \frac{\sqrt{n^5-8} - n\sqrt {n\left(n^2+5\right)}} {\sqrt {n} }$	4-6	$\lim_{n\to \infty } \left( \sqrt {n^2-3n+2} -n \right)$
4-7	$\lim_{n\to \infty } \left(n+\sqrt[3]{4-n^3}\right)$	4-8	$\lim_{n\to \infty} \left(\sqrt{n\left(n+2 \right)} -\sqrt{n^2-2n+3}\right)$
4-9	$\lim_{n\to \infty} \left(\sqrt{(n+2)(n+1)} - \sqrt{(n - 1)(n+3)} \right)$	4-10	$\lim_{n\to \infty } n^2\left({\sqrt {n\left( {n^4-1} \right)} -\sqrt {n^5-8} } \right)$
4-11	$\lim_{n\to \infty} n\left(\sqrt[3]{5+8n^3}-2n\right)$	4-12	$\lim_{n\to \infty} n^2\left(\sqrt[3]{5+n^3}-\sqrt[3]{3+n^3} \right)$
4-13	$\lim_{n\to \infty} \left(\sqrt[3]{(n+2)^2}-\sqrt[3]{(n-3)^2}\right)$	4-14	$\lim_{n\to \infty} \frac{\sqrt{(n+1)^3} -\sqrt{n(n-1)(n-3)}} {\sqrt {n}}$
4-15	$\lim_{n\to \infty} \left(\sqrt{n^2+3n-2} - \sqrt {n^2-3}\right)$	4-16	$\lim_{n\to \infty} \sqrt{n} \left(\sqrt{n+2} - \sqrt{n-3} \right)$
4-17	$\lim_{n\to \infty} \frac{\sqrt{n\left(n^5+9\right)} -\sqrt{\left(n^4-1\right)\left({n^2+5}\right)}} {n}$	4-18	$\lim_{n\to \infty} \left(\sqrt{n(n+5)} -n \right)$
4-19	$\lim_{n\to \infty} \sqrt{n^3+8} \left(\sqrt {n^3+2} - \sqrt{n^3-1}\right)$	4-20	$\lim_{n\to \infty} \frac{\sqrt{\left(n^3+1 \right)\left(n^2+3 \right)} -\sqrt{n\left(n^4+2 \right)}} {2\sqrt{n}}$
4-21	$\lim_{n\to \infty} \left(\sqrt{\left(n^2+1 \right)\left(n^2+2 \right)} -\sqrt{\left(n^2-1 \right)\left(n^2-2 \right)} \right)$	4-22	$\lim_{n\to \infty} \frac{\sqrt{\left(n^5+1 \right)\left(n^2-1\right)} - n\sqrt{n\left(n^4+1 \right)}} {n}$
4-23	$\lim_{n\to \infty} \frac{\sqrt{\left(n^4+1 \right)\left(n^2-1 \right)} - \sqrt{n^6-1}} {n}$	4-24	$\lim_{n\to \infty} \left( n - \sqrt{n(n-1)}\right)$
4-25	$\lim_{n\to \infty} n^3\left(\sqrt[3]{n^2\left(n^6+4 \right)} - \sqrt[3]{n^8-1} \right)$	4-26	$\lim_{n\to \infty} \left(n \sqrt{n} - \sqrt{n(n+1)(n+2)}\right)$
4-27	$\lim_{n\to \infty} \sqrt[3]{n}\left(\sqrt[3]{n^2} - \sqrt[3]{n(n-1)}\right)$	4-28	$\lim_{n\to \infty} \sqrt{n+2} \left(\sqrt{n+3} -\sqrt{n-4}\right)$
4-29	$\lim_{n\to \infty} n\left(\sqrt{n^4+3} - \sqrt {n^4-2}\right)$	4-30	$\lim_{n\to \infty} \sqrt{n(n+1)(n+2)} \left(\sqrt{n^3-3} - \sqrt {n^3-2}\right)$
4-31	$\lim_{n\to \infty} \frac{\sqrt{\left(n^2+5 \right)\left(n^4+2 \right)} - \sqrt{n^6-3n^3+5}} {n}$