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

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

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

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

- \infty

Solution steps

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

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

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

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t an g e n t f (x) = x^{2} + 2, (- 4, 18)

\frac{d}{d x} (sec^{2} (x) + csc^{2} (x))

\frac{d}{d x} (g (h, x) (h (x)))

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