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

解答

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

\infty

求解步骤

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

化简

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

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

改写以应用洛必达法则

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

使用洛必达法则

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

化简

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

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

使用极限链式法则:

\infty

= \infty

x \to 2 lim (6 x^{2} + 3 x - 9)

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

x \to 3 lim (x^{2} + 7 x + 2)

x \to 0 lim (\frac{ln ( 1 + m x )}{x})

m \to 0 lim (\frac{sin ( m )}{m})