次の数列の極限を求めます。 (1) $\lim_{n \to \infty} (3 + \frac{2}{n})$ (2) $\lim_{n \to \infty} \frac{4n - 3}{3n + 1}$ (3) $\lim_{n \to \infty} \frac{2n^2 + 3n - 1}{3n^2 - n + 5}$ (4) $\lim_{n \to \infty} (\sqrt{n^2 - 2n - 1} - n)$ (5) $\lim_{n \to \infty} (\sqrt{n + 1} - \sqrt{n})$ (6) $\lim_{n \to \infty} (n - n^2)$ (7) $\lim_{n \to \infty} \frac{3^n}{3^n + 2^n}$ (8) $\lim_{n \to \infty} (1 + \frac{1}{n})^n$ (9) $\lim_{n \to \infty} (1 + \frac{1}{3n})^n$ (10) $\lim_{n \to \infty} \frac{1 + 2 + \cdots + n}{n^2}$ (11) $\lim_{n \to \infty} \frac{1^2 + 2^2 + \cdots + n^2}{n^3}$ (12) $\lim_{n \to \infty} \frac{\sin n}{n}$

はい、承知いたしました。画像の問題を解いていきます。

1. 問題の内容

次の数列の極限を求めます。

(1) $\lim_{n \to \infty} (3 + \frac{2}{n})$

(2) $\lim_{n \to \infty} \frac{4n - 3}{3n + 1}$

(3) $\lim_{n \to \infty} \frac{2n^2 + 3n - 1}{3n^2 - n + 5}$

(4) $\lim_{n \to \infty} (\sqrt{n^2 - 2n - 1} - n)$

(5) $\lim_{n \to \infty} (\sqrt{n + 1} - \sqrt{n})$

(6) $\lim_{n \to \infty} (n - n^2)$

(7) $\lim_{n \to \infty} \frac{3^n}{3^n + 2^n}$

(8) $\lim_{n \to \infty} (1 + \frac{1}{n})^n$

(9) $\lim_{n \to \infty} (1 + \frac{1}{3n})^n$

(10) $\lim_{n \to \infty} \frac{1 + 2 + \cdots + n}{n^2}$

(11) $\lim_{n \to \infty} \frac{1^2 + 2^2 + \cdots + n^2}{n^3}$

(12) $\lim_{n \to \infty} \frac{\sin n}{n}$

2. 解き方の手順

(1)

$\lim_{n \to \infty} (3 + \frac{2}{n}) = 3 + \lim_{n \to \infty} \frac{2}{n} = 3 + 0 = 3$

(2)

$\lim_{n \to \infty} \frac{4n - 3}{3n + 1} = \lim_{n \to \infty} \frac{n(4 - \frac{3}{n})}{n(3 + \frac{1}{n})} = \lim_{n \to \infty} \frac{4 - \frac{3}{n}}{3 + \frac{1}{n}} = \frac{4 - 0}{3 + 0} = \frac{4}{3}$

(3)

$\lim_{n \to \infty} \frac{2n^2 + 3n - 1}{3n^2 - n + 5} = \lim_{n \to \infty} \frac{n^2(2 + \frac{3}{n} - \frac{1}{n^2})}{n^2(3 - \frac{1}{n} + \frac{5}{n^2})} = \lim_{n \to \infty} \frac{2 + \frac{3}{n} - \frac{1}{n^2}}{3 - \frac{1}{n} + \frac{5}{n^2}} = \frac{2 + 0 - 0}{3 - 0 + 0} = \frac{2}{3}$

(4)

$\lim_{n \to \infty} (\sqrt{n^2 - 2n - 1} - n) = \lim_{n \to \infty} \frac{(\sqrt{n^2 - 2n - 1} - n)(\sqrt{n^2 - 2n - 1} + n)}{\sqrt{n^2 - 2n - 1} + n} = \lim_{n \to \infty} \frac{n^2 - 2n - 1 - n^2}{\sqrt{n^2 - 2n - 1} + n} = \lim_{n \to \infty} \frac{-2n - 1}{\sqrt{n^2 - 2n - 1} + n} = \lim_{n \to \infty} \frac{n(-2 - \frac{1}{n})}{\sqrt{n^2(1 - \frac{2}{n} - \frac{1}{n^2})} + n} = \lim_{n \to \infty} \frac{n(-2 - \frac{1}{n})}{n\sqrt{1 - \frac{2}{n} - \frac{1}{n^2}} + n} = \lim_{n \to \infty} \frac{-2 - \frac{1}{n}}{\sqrt{1 - \frac{2}{n} - \frac{1}{n^2}} + 1} = \frac{-2 - 0}{\sqrt{1 - 0 - 0} + 1} = \frac{-2}{1 + 1} = -1$

(5)

$\lim_{n \to \infty} (\sqrt{n + 1} - \sqrt{n}) = \lim_{n \to \infty} \frac{(\sqrt{n + 1} - \sqrt{n})(\sqrt{n + 1} + \sqrt{n})}{\sqrt{n + 1} + \sqrt{n}} = \lim_{n \to \infty} \frac{n + 1 - n}{\sqrt{n + 1} + \sqrt{n}} = \lim_{n \to \infty} \frac{1}{\sqrt{n + 1} + \sqrt{n}} = 0$

(6)

$\lim_{n \to \infty} (n - n^2) = \lim_{n \to \infty} n^2(\frac{1}{n} - 1) = -\infty$

(7)

$\lim_{n \to \infty} \frac{3^n}{3^n + 2^n} = \lim_{n \to \infty} \frac{3^n}{3^n(1 + (\frac{2}{3})^n)} = \lim_{n \to \infty} \frac{1}{1 + (\frac{2}{3})^n} = \frac{1}{1 + 0} = 1$

(8)

$\lim_{n \to \infty} (1 + \frac{1}{n})^n = e$

(9)

$\lim_{n \to \infty} (1 + \frac{1}{3n})^n = \lim_{n \to \infty} [(1 + \frac{1}{3n})^{3n}]^{\frac{1}{3}} = e^{\frac{1}{3}}$

(10)

$\lim_{n \to \infty} \frac{1 + 2 + \cdots + n}{n^2} = \lim_{n \to \infty} \frac{\frac{n(n + 1)}{2}}{n^2} = \lim_{n \to \infty} \frac{n^2 + n}{2n^2} = \lim_{n \to \infty} \frac{1 + \frac{1}{n}}{2} = \frac{1 + 0}{2} = \frac{1}{2}$

(11)

$\lim_{n \to \infty} \frac{1^2 + 2^2 + \cdots + n^2}{n^3} = \lim_{n \to \infty} \frac{\frac{n(n + 1)(2n + 1)}{6}}{n^3} = \lim_{n \to \infty} \frac{2n^3 + 3n^2 + n}{6n^3} = \lim_{n \to \infty} \frac{2 + \frac{3}{n} + \frac{1}{n^2}}{6} = \frac{2}{6} = \frac{1}{3}$

(12)

$-\frac{1}{n} \le \frac{\sin n}{n} \le \frac{1}{n}$

$\lim_{n \to \infty} -\frac{1}{n} = 0$

$\lim_{n \to \infty} \frac{1}{n} = 0$

よって、$\lim_{n \to \infty} \frac{\sin n}{n} = 0$ （はさみうちの原理）

3. 最終的な答え

(1) 3

(2) 4/3

(3) 2/3

(4) -1

(5) 0

(6) $-\infty$

(7) 1

(8) e

(9) $e^{\frac{1}{3}}$

(10) 1/2

(11) 1/3

(12) 0