What is the limit as x approaches infinity of ln(x^{2/x}) ?

The limit as x approaches infinity of ln(x^{2/x}) is 0

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limit as x approaches infinity of ln(x^{2/x})

x \to \infty lim (ln (x^{\frac{2}{x}}))

0

Solution steps

x \to \infty lim (ln (x^{\frac{2}{x}}))

ln (x^{\frac{2}{x}}) = \frac{2}{x} ln (x)

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

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

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

Simplify

\frac{1}{x} ln (x) : \frac{ln ( x )}{x}

= 2 \cdot x \to \infty lim (\frac{ln ( x )}{x})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{\frac{1}{x}}{1})

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

Simplify

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

Apply Infinity Property:

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

= 2 \cdot 0

Simplify

= 0

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

x \to 1 + lim (ln (x) ln (x - 1))

x \to 4 lim (3 (x - 4)^{x - 4})

x \to - 3 - lim (\frac{4 - x ^{2}}{x + 3})

x \to 0 lim (2 x csc (x))

What is the limit as x approaches infinity of ln(x^{2/x}) ?
The limit as x approaches infinity of ln(x^{2/x}) is 0