What is the limit as x approaches 0+of 2xln(1/x) ?

The limit as x approaches 0+of 2xln(1/x) is 0

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limit as x approaches 0+of 2xln(1/x)

x \to 0 + lim (2 x ln (\frac{1}{x}))

0

Solution steps

x \to 0 + lim (2 x ln (\frac{1}{x}))

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

= 2 \cdot x \to 0 + lim (x ln (\frac{1}{x}))

Rewrite for L'Hopital

= 2 \cdot 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}}})

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

Simplify

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

= 2 \cdot x \to 0 + lim (x)

Plug in the value

x = 0

= 2 \cdot 0

Simplify

= 0

x \to \frac{1}{4} lim (ln (x))

x \to 0 lim (1 - 5 x^{3})

x \to 0 lim (\frac{x ^{4} - 2}{x ^{3}})

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

x \to 4 lim (4 x^{3} - 3 x^{2} + 1)

What is the limit as x approaches 0+of 2xln(1/x) ?
The limit as x approaches 0+of 2xln(1/x) is 0