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人気のある微分積分 >

limit as x approaches 0+of x^2*e^{1/x}

解

x \to 0 + lim (x^{2} \cdot e^{\frac{1}{x}})

解

\infty

解答ステップ

x \to 0 + lim (x^{2} e^{\frac{1}{x}})

ロピタル向けに書き換える

= x \to 0 + lim (\frac{e ^{\frac{1}{x}}}{\frac{1}{x ^{2}}})

ロピタルの定理を適用する

= x \to 0 + lim \frac{- \frac{e ^{\frac{1}{x}}}{x ^{2}}}{- \frac{2}{x ^{3}}}

簡素化

\frac{- \frac{e ^{\frac{1}{x}}}{x ^{2}}}{- \frac{2}{x ^{3}}} : \frac{1}{2} e^{\frac{1}{x}} x

= x \to 0 + lim (\frac{1}{2} e^{\frac{1}{x}} x)

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

= \frac{1}{2} \cdot x \to 0 + lim (e^{\frac{1}{x}} x)

ロピタル向けに書き換える

= \frac{1}{2} \cdot x \to 0 + lim (\frac{e ^{\frac{1}{x}}}{\frac{1}{x}})

ロピタルの定理を適用する

= \frac{1}{2} \cdot x \to 0 + lim \frac{- \frac{e ^{\frac{1}{x}}}{x ^{2}}}{- \frac{1}{x ^{2}}}

簡素化

\frac{- \frac{e ^{\frac{1}{x}}}{x ^{2}}}{- \frac{1}{x ^{2}}} : e^{\frac{1}{x}}

= \frac{1}{2} \cdot x \to 0 + lim (e^{\frac{1}{x}})

連鎖法則の制限を適用する:

\infty

= \frac{1}{2} \cdot \infty

無限大の特性を適用する:

c \cdot \infty = \infty

= \infty

グラフ

人気の例

limit as x approaches 0 of x^2-x+1

x \to 0 lim (x^{2} - x + 1)

limit as t approaches 0+of tln(t)

t \to 0 + lim (t ln (t))

limit as x approaches 2 of (x-3)/(x+1)

x \to 2 lim (\frac{x - 3}{x + 1})

limit as x approaches-2 of F(x)

x \to - 2 lim (F (x))

limit as x approaches-1 of x^{12}

x \to - 1 lim (x^{12})