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 -infinity

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

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

- \infty

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})

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

With the exception of indeterminate form

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

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

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

= \infty \cdot (- \infty)

Apply Infinity Property:

\infty (- \infty) = - \infty

= - \infty

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

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

x \to - 2 + lim (\frac{1}{( x + 2 ) ^{5}})

x \to \infty lim (45)

x \to 1 lim (x \frac{1}{1 - x})

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 -infinity