What is the limit as x approaches 0+of x(ln(x))^2 ?

The limit as x approaches 0+of x(ln(x))^2 is 0

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limit as x approaches 0+of x(ln(x))^2

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

0

Solution steps

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

Rewrite for L'Hopital

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

Apply L'Hopital's Rule

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

Simplify

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

= x \to 0 + lim (- 2 x ln (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 (x))

Rewrite for L'Hopital

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

Apply L'Hopital's Rule

= - 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 (- 0)

Simplify

= 0

What is the limit as x approaches 0+of x(ln(x))^2 ?
The limit as x approaches 0+of x(ln(x))^2 is 0