limit as x approaches infinity of ((e^{(15)/x}-1))/((15)/x)

解答

x \to \infty lim \frac{( e ^{\frac{15}{x}} - 1 )}{\frac{15}{x}}

1

求解步骤

x \to \infty lim (\frac{e ^{\frac{15}{x}} - 1}{\frac{15}{x}})

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

= \frac{1}{15} \cdot x \to \infty lim (\frac{e ^{\frac{15}{x}} - 1}{\frac{1}{x}})

化简

= \frac{1}{15} \cdot x \to \infty lim (x (e^{\frac{15}{x}} - 1))

改写以应用洛必达法则

= \frac{1}{15} \cdot x \to \infty lim (\frac{- 1 + e ^{\frac{15}{x}}}{\frac{1}{x}})

使用洛必达法则

= \frac{1}{15} \cdot x \to \infty lim \frac{- \frac{15 e ^{\frac{15}{x}}}{x ^{2}}}{- \frac{1}{x ^{2}}}

化简

\frac{- \frac{15 e ^{\frac{15}{x}}}{x ^{2}}}{- \frac{1}{x ^{2}}} : 15 e^{\frac{15}{x}}

= \frac{1}{15} \cdot x \to \infty lim (15 e^{\frac{15}{x}})

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

= \frac{1}{15} \cdot 15 \cdot x \to \infty lim (e^{\frac{15}{x}})

使用极限链式法则:

1

= \frac{1}{15} \cdot 15 \cdot 1

化简

\frac{1}{15} \cdot 15 \cdot 1 : 1

= 1

in v erse l a pl a ce (s + \frac{3}{2})^{2}

\int e^{\frac{- x}{k}} \cdot \frac{- x}{k} d x

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\frac{d y}{d x} = x y + y^{2}