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

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

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limit as x approaches infinity of (ln(x))/(e^x)

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

0

Solution steps

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

Apply L'Hopital's Rule

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

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= x \to \infty lim (\frac{1}{e ^{x} 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 \infty} ( 1 )}{lim _{x \to \infty} ( e ^{x} x )}

x \to \infty lim (1) = 1

x \to \infty lim (e^{x} x) = \infty

= \frac{1}{\infty}

Apply Infinity Property:

\frac{c}{\infty} = 0

= 0

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