What is the limit as n approaches infinity of (ln(1/n))/(sqrt(2n)) ?

The limit as n approaches infinity of (ln(1/n))/(sqrt(2n)) is 0

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limit as n approaches infinity of (ln(1/n))/(sqrt(2n))

n \to \infty lim (\frac{ln ( \frac{1}{n} )}{2 n})

0

Solution steps

n \to \infty lim (\frac{ln ( \frac{1}{n} )}{2 n})

Simplify

\frac{ln ( \frac{1}{n} )}{2 n} : - \frac{ln ( n )}{2 n}

= n \to \infty lim (- \frac{ln ( n )}{2 n})

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= - \frac{1}{2} \cdot n \to \infty lim (\frac{ln ( n )}{n})

Apply L'Hopital's Rule

= - \frac{1}{2} \cdot n \to \infty lim (\frac{\frac{1}{n}}{\frac{1}{2 n}})

Simplify

\frac{\frac{1}{n}}{\frac{1}{2 n}} : \frac{2}{n}

= - \frac{1}{2} \cdot n \to \infty lim (\frac{2}{n})

Apply Infinity Property:

x \to \infty lim (\frac{c}{x ^{a}}) = 0

= - \frac{1}{2} \cdot 0

Simplify

= 0

What is the limit as n approaches infinity of (ln(1/n))/(sqrt(2n)) ?
The limit as n approaches infinity of (ln(1/n))/(sqrt(2n)) is 0