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人気のある微分積分 >

limit as x approaches infinity of ((x^p)/(e^x))^{1/n}

解

x \to \infty lim ((\frac{x ^{p}}{e ^{x}})^{\frac{1}{n}})

解

0

解答ステップ

x \to \infty lim ((\frac{x ^{p}}{e ^{x}})^{\frac{1}{n}})

x \to a lim [f (x)]^{b} = [x \to a lim f (x)]^{b}

不定形式の場合を除く

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

指数の規則を適用する:

\frac{1}{a ^{b}} = a^{- b}

\frac{x ^{p}}{e ^{x}} = (x^{p}) e^{- x}

= (x \to \infty lim (x^{p} e^{- x}))^{\frac{1}{n}}

ロピタル向けに書き換える

= (x \to \infty lim (\frac{x ^{p}}{\frac{1}{e ^{- x}}}))^{\frac{1}{n}}

ロピタルの定理を適用する

= (x \to \infty lim (\frac{p x ^{p - 1}}{e ^{x}}))^{\frac{1}{n}}

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

= (p \cdot x \to \infty lim (\frac{x ^{p - 1}}{e ^{x}}))^{\frac{1}{n}}

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

= (p \frac{lim _{x \to \infty} ( x ^{p - 1} )}{lim _{x \to \infty} ( e ^{x} )})^{\frac{1}{n}}

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

未定義

x \to \infty lim (e^{x}) = \infty

= (p \frac{未定義}{\infty})^{\frac{1}{n}}

簡素化

(p \frac{未定義}{\infty})^{\frac{1}{n}} : 0

= 0

グラフ

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