limit as x approaches infinity of \sqrt[x]{rx}

解

x \to \infty lim (x r x)

1

解答ステップ

x \to \infty lim (x r x)

指数の規則を適用する:

(a \cdot b)^{n} = a^{n} \cdot b^{n}

x r x = (r)^{\frac{1}{x}} (x)^{\frac{1}{x}}

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

x \to a lim [f (x) \cdot g (x)] = x \to a lim f (x) \cdot x \to a lim g (x)

不定形式の場合を除く

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

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

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

= 1 \cdot 1

簡素化

= 1