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

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

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

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

0

Solution steps

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

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

With the exception of indeterminate form

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

x \to 0 + lim (1) = 1

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

= \frac{1}{\infty}

Apply Infinity Property:

\frac{c}{\infty} = 0

= 0

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