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人気のある微分積分 >

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

解

x \to 0 + lim (x^{2} 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-infinity of x+4

x \to - \infty lim (x + 4)

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

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

limit as x approaches 0 of (x+cos(2x))^{csc(3x)}

x \to 0 lim ((x + cos (2 x))^{c s c (3 x)})

limit as x approaches 0 of cos(x-1)

x \to 0 lim (cos (x - 1))

limit as x approaches 0 of (x^2-3x)/(3x)

x \to 0 lim (\frac{x ^{2} - 3 x}{3 x})