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

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

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

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

0

Solution steps

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

Apply L'Hopital's Rule

= x \to \infty lim \frac{1}{ln ^{x} ( x ) ( ln ( ln ( x ) ) + \frac{1}{l n ( x )} )}

Simplify

\frac{1}{ln ^{x} ( x ) ( ln ( ln ( x ) ) + \frac{1}{l n ( x )} )} : \frac{ln ( x )}{ln ^{x} ( x ) ( ln ( ln ( x ) ) ln ( x ) + 1 )}

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

Apply L'Hopital's Rule

= x \to \infty lim (\frac{\frac{1}{x}}{ln ^{2} ( ln ( x ) ) ln ^{1 + x} ( x ) + 2 ln ( ln ( x ) ) ln ^{x} ( x ) + ln ^{x - 1} ( x ) + \frac{l n ^{x} ( x ) ( l n ( l n ( x ) ) + 1 )}{x}})

Simplify

\frac{\frac{1}{x}}{ln ^{2} ( ln ( x ) ) ln ^{1 + x} ( x ) + 2 ln ( ln ( x ) ) ln ^{x} ( x ) + ln ^{x - 1} ( x ) + \frac{l n ^{x} ( x ) ( l n ( l n ( x ) ) + 1 )}{x}} : \frac{1}{x ln ^{2} ( ln ( x ) ) ln ^{1 + x} ( x ) + 2 x ln ( ln ( x ) ) ln ^{x} ( x ) + x ln ^{x - 1} ( x ) + ln ^{x} ( x ) ( ln ( ln ( x ) ) + 1 )}

= x \to \infty lim (\frac{1}{x ln ^{2} ( ln ( x ) ) ln ^{1 + x} ( x ) + 2 x ln ( ln ( x ) ) ln ^{x} ( x ) + x ln ^{x - 1} ( x ) + ln ^{x} ( x ) ( ln ( ln ( 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 \infty} ( 1 )}{lim _{x \to \infty} ( x ln ^{2} ( ln ( x ) ) ln ^{1 + x} ( x ) + 2 x ln ( ln ( x ) ) ln ^{x} ( x ) + x ln ^{x - 1} ( x ) + ln ^{x} ( x ) ( ln ( ln ( x ) ) + 1 ) )}

x \to \infty lim (1) = 1

x \to \infty lim (x ln^{2} (ln (x)) ln^{1 + x} (x) + 2 x ln (ln (x)) ln^{x} (x) + x ln^{x - 1} (x) + ln^{x} (x) (ln (ln (x)) + 1)) = \infty

= \frac{1}{\infty}

Apply Infinity Property:

\frac{c}{\infty} = 0

= 0

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