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

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

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

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

\infty

Solution steps

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

Rewrite for L'Hopital

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

Apply L'Hopital's Rule

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

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

Simplify

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

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

Apply the Limit Chain Rule:

\infty

= \infty

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