What is the limit as x approaches 0-of (e^{1/x})/x ?

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

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

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

0

Solution steps

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

\frac{e ^{\frac{1}{x}}}{x} = e^{\frac{1}{x}} \frac{1}{x}

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

Rewrite for L'Hopital

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

Apply L'Hopital's Rule

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

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

Simplify

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

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

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

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

Apply the Limit Chain Rule:

0

= - 0

Simplify

= 0

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What is the limit as x approaches 0-of (e^{1/x})/x ?
The limit as x approaches 0-of (e^{1/x})/x is 0