What is the limit as x approaches 0+of 3xln(1/x) ?

The limit as x approaches 0+of 3xln(1/x) is 0

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limit as x approaches 0+of 3xln(1/x)

x \to 0 + lim (3 x ln (\frac{1}{x}))

0

Solution steps

x \to 0 + lim (3 x ln (\frac{1}{x}))

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

= 3 \cdot x \to 0 + lim (x ln (\frac{1}{x}))

Rewrite for L'Hopital

= 3 \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}}})

= 3 \cdot x \to 0 + lim (\frac{- \frac{1}{x}}{- \frac{1}{x ^{2}}})

Simplify

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

= 3 \cdot x \to 0 + lim (x)

Plug in the value

x = 0

= 3 \cdot 0

Simplify

= 0

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

x \to 0 lim (\frac{4 x}{e ^{x} - 1})

x \to 1 lim (x^{2} - 4 x + 5)

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

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

What is the limit as x approaches 0+of 3xln(1/x) ?
The limit as x approaches 0+of 3xln(1/x) is 0