What is the limit as x approaches 0+of ln((1/x)^x) ?

The limit as x approaches 0+of ln((1/x)^x) is 0

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limit as x approaches 0+of ln((1/x)^x)

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

0

Solution steps

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

ln ((\frac{1}{x})^{x}) = x ln ((\frac{1}{x}))

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

Rewrite for L'Hopital

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

Apply L'Hopital's Rule

: x \to 0 + lim (\frac{- \frac{1}{x}}{- \frac{1}{x ^{2}}})

= x \to 0 + lim (\frac{- \frac{1}{x}}{- \frac{1}{x ^{2}}})

Simplify

\frac{- \frac{1}{x}}{- \frac{1}{x ^{2}}} : x

= x \to 0 + lim (x)

Plug in the value

x = 0

= 0

x \to \infty lim (- x^{- 2})

x \to 0 + lim (x x)

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

h \to 0 + lim (h)

x \to 5 lim (sin (x))

What is the limit as x approaches 0+of ln((1/x)^x) ?
The limit as x approaches 0+of ln((1/x)^x) is 0