Математика
★ Рубрика: Математика
★ Тема: примеры

Примеры вычисления пределов дробей

Здесь предлагаются решения однотипных примеров на нахождение пределов рациональных дробей. Примеры решаются выполнением преобразований и определением степени многочленов в числителе и знаменателе.

1.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{\left( {3 - n} \right)}^2} + {{\left( {3 + n} \right)}^2}}}{{{{\left( {3 - n} \right)}^2} - {{\left( {3 + n} \right)}^2}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{2{n^2} + o({n^2})}}{{o({n^2})}} = \infty \]
2.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{\left( {3 - n} \right)}^4} - {{\left( {2 - n} \right)}^4}}}{{{{\left( {1 - n} \right)}^4} - {{\left( {1 + n} \right)}^4}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{ - 4{n^3} + o({n^3})}}{{ - 8{n^3} + o({n^3})}} = \frac{1}{2}\]
3.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{\left( {3 - n} \right)}^4} - {{\left( {2 - n} \right)}^4}}}{{{{\left( {1 - n} \right)}^3} - {{\left( {1 + n} \right)}^3}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{ - 4{n^3} + o({n^3})}}{{ - 2{n^3} + o({n^3})}} = 2\]
4.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{\left( {1 - n} \right)}^4} - {{\left( {1 + n} \right)}^4}}}{{{{\left( {1 + n} \right)}^3} - {{\left( {1 - n} \right)}^3}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{ - 8{n^3} + o({n^3})}}{{2{n^3} + o({n^3})}} =  - 4\]
5.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{\left( {6 - n} \right)}^2} - {{\left( {6 + n} \right)}^2}}}{{{{\left( {6 + n} \right)}^2} - {{\left( {1 - n} \right)}^2}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{ - 24n}}{{14n + o(n)}} =  - \frac{8}{7}\]
6.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{\left( {n + 1} \right)}^3} - {{\left( {n + 1} \right)}^2}}}{{{{\left( {n - 1} \right)}^3} - {{\left( {n + 1} \right)}^3}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{{n^3} + o({n^3})}}{{o({n^3})}} = \infty \]
7.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{\left( {1 + 2n} \right)}^3} - 8{n^3}}}{{{{\left( {1 + 2n} \right)}^3} + 4{n^2}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{o({n^3})}}{{8{n^3} + o({n^3})}} = 0\]
8.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{\left( {3 - 4n} \right)}^2}}}{{{{\left( {n - 3} \right)}^3} - {{\left( {n + 3} \right)}^3}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{16{n^2} + o({n^2})}}{{ - 18{n^2} + o({n^2})}} =  - \frac{8}{9}\]
9.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{\left( {3 - n} \right)}^3}}}{{{{\left( {n + 1} \right)}^2} - {{\left( {n + 1} \right)}^3}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{ - {n^3} + o({n^3})}}{{ - {n^3} + o({n^3})}} = 1\]
10.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{\left( {n + 1} \right)}^2} + {{(n - 1)}^2} - {{\left( {n + 2} \right)}^3}}}{{{{\left( {4 - n} \right)}^3}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{ - {n^3} + o({n^3})}}{{ - {n^3} + o({n^3})}} = 1\]
11.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{2{{(n + 1)}^3} - {{\left( {n - 2} \right)}^3}}}{{{n^2} + 2n - 3}} = \mathop {\lim }\limits_{n \to \infty } \frac{{{n^3} + o({n^3})}}{{o({n^3})}} = \infty \]
12.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{(n + 1)}^3} + {{\left( {n + 2} \right)}^3}}}{{{{\left( {n + 4} \right)}^3} + {{\left( {n + 5} \right)}^3}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{2{n^3} + o({n^3})}}{{2{n^3} + o({n^3})}} = 1\]
13.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{(n + 3)}^3} + {{\left( {n + 4} \right)}^3}}}{{{{\left( {n + 3} \right)}^4} - {{\left( {n + 4} \right)}^4}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{2{n^3} + o({n^3})}}{{ - 4{n^3} + o({n^3})}} =  - \frac{1}{2}\]
14.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{(n + 1)}^4} - {{\left( {n - 1} \right)}^4}}}{{{{\left( {n + 1} \right)}^3} + {{\left( {n - 1} \right)}^3}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{8{n^3} + o({n^3})}}{{2{n^3} + o({n^3})}} = 4\]
15.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{8{n^3} - 2n}}{{{{\left( {n + 1} \right)}^4} - {{\left( {n - 1} \right)}^4}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{8{n^3} + o({n^3})}}{{8{n^3} + o({n^3})}} = 1\]
16.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{(n + 6)}^3} - {{\left( {n + 1} \right)}^3}}}{{{{\left( {2n + 3} \right)}^2} + {{\left( {n + 4} \right)}^2}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{15{n^2} + o({n^2})}}{{5{n^2} + o({n^2})}} = 3\]
17.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{(2n - 3)}^3} - {{\left( {n + 5} \right)}^3}}}{{{{\left( {3n - 1} \right)}^3} + {{\left( {2n + 3} \right)}^3}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{7{n^3} + o({n^3})}}{{35{n^3} + o({n^3})}} = \frac{1}{5}\]
18.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{(n + 10)}^2} + {{\left( {3n + 1} \right)}^2}}}{{{{\left( {n + 6} \right)}^3} - {{\left( {n + 1} \right)}^3}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{10{n^2} + o({n^2})}}{{15{n^2} + o({n^2})}} = \frac{2}{3}\]
19.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{(2n + 1)}^3} + {{\left( {3n + 2} \right)}^3}}}{{{{\left( {2n + 3} \right)}^3} - {{\left( {n - 7} \right)}^3}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{35{n^3} + o({n^3})}}{{7{n^3} + o({n^3})}} = 5\]
20.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{(n + 7)}^3} - {{\left( {n + 2} \right)}^3}}}{{{{\left( {3n + 2} \right)}^2} + {{\left( {4n + 1} \right)}^2}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{15{n^2} + o({n^2})}}{{25{n^2} + o({n^2})}} = \frac{3}{5}\]
21.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{(2n + 1)}^3} - {{\left( {2n + 3} \right)}^3}}}{{{{\left( {2n + 1} \right)}^2} + {{\left( {2n + 3} \right)}^2}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{ - 24{n^2} + o({n^2})}}{{8{n^2} + o({n^2})}} =  - 3\]
22.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{n^3} - {{\left( {n - 1} \right)}^3}}}{{{{\left( {n + 1} \right)}^4} - {n^4}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{o({n^3})}}{{4{n^3} + o({n^3})}} = 0\]
23.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{(n + 2)}^4} - {{\left( {n - 2} \right)}^4}}}{{{{\left( {n + 5} \right)}^2} + {{\left( {n - 5} \right)}^2}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{16{n^3} + o({n^3})}}{{o({n^3})}} = \infty \]
24.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{(n + 1)}^4} - {{\left( {n - 1} \right)}^4}}}{{{{\left( {n + 1} \right)}^3} + {{\left( {n - 1} \right)}^3}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{8{n^3} + o({n^3})}}{{2{n^3} + o({n^3})}} = 4\]
25.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{(n + 1)}^3} - {{\left( {n - 1} \right)}^3}}}{{{{\left( {n + 1} \right)}^2} - {{\left( {n - 1} \right)}^2}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{6{n^2} + o({n^2})}}{{o({n^2})}} = \infty \]
26.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{(n + 1)}^3} - {{\left( {n - 1} \right)}^3}}}{{{{\left( {n + 1} \right)}^2} + {{\left( {n - 1} \right)}^2}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{6{n^2} + o({n^2})}}{{2{n^2} + o({n^2})}} = 3\]
27.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{(n + 2)}^3} - {{\left( {n - 2} \right)}^3}}}{{{n^4} + 2{n^2} - 1}} = \mathop {\lim }\limits_{n \to \infty } \frac{{o({n^4})}}{{{n^4} + o({n^4})}} = 0\]
28.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{(n + 1)}^3} + {{\left( {n - 1} \right)}^3}}}{{{n^3} - 3n}} = \mathop {\lim }\limits_{n \to \infty } \frac{{2{n^3} + o({n^3})}}{{{n^3} + o({n^3})}} = 2\]
 
29.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{(n + 1)}^3} + {{\left( {n - 1} \right)}^3}}}{{{n^3} + 1}} = \mathop {\lim }\limits_{n \to \infty } \frac{{2{n^3} + o({n^3})}}{{{n^3} + o({n^3})}} = 2\]
30.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{(n + 2)}^2} - {{\left( {n - 2} \right)}^2}}}{{{{\left( {n + 3} \right)}^2}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{o({n^2})}}{{{n^2} + o({n^2})}} = 0\]
31.
   \[\mathop {\lim }\limits_{n \to \infty } \frac{{{{(2n + 1)}^2} - {{\left( {n + 1} \right)}^2}}}{{{n^2} + n + 1}} = \mathop {\lim }\limits_{n \to \infty } \frac{{3{n^2} + o({n^2})}}{{{n^2} + o({n^2})}} = 3\]
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