limit as x approaches infinity of (sqrt(1+\sqrt{1/x)}-1)sqrt(x)

解

x \to \infty lim 1 + \frac{1}{x} - 1 x

\frac{1}{2}

十進法表記

0.5

解答ステップ

x \to \infty lim 1 + \frac{1}{x} - 1 x

1 + \frac{1}{x} - 1

の共役で乗じる

\frac{\frac{1}{x}}{1 + \frac{1}{x} + 1}

= x \to \infty lim \frac{\frac{1}{x}}{1 + \frac{1}{x} + 1} x

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= x \to \infty lim \frac{\frac{1}{x} x}{1 + \frac{1}{x} + 1}

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

不定形式の場合を除く

= \frac{lim _{x \to \infty} ( \frac{1}{x} x )}{lim _{x \to \infty} ( 1 + \frac{1}{x} + 1 )}

x \to \infty lim (\frac{1}{x} x) = 1

x \to \infty lim 1 + \frac{1}{x} + 1 = 2

= \frac{1}{2}