What is the limit as x approaches 0+of (ln(x))/(e^x) ?

The limit as x approaches 0+of (ln(x))/(e^x) is -infinity

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limit as x approaches 0+of (ln(x))/(e^x)

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

- \infty

Solution steps

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

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 +} ( ln ( x ) )}{lim _{x \to 0 +} ( e ^{x} )}

x \to 0 + lim (ln (x)) = - \infty

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

= \frac{- \infty}{1}

Apply Infinity Property:

\frac{- \infty}{c} = - \infty

= - \infty

What is the limit as x approaches 0+of (ln(x))/(e^x) ?
The limit as x approaches 0+of (ln(x))/(e^x) is -infinity