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

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