What is the limit as x approaches 0+of x^9(ln(x))^2 ?

The limit as x approaches 0+of x^9(ln(x))^2 is 0

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limit as x approaches 0+of x^9(ln(x))^2

x \to 0 + lim (x^{9} (ln (x))^{2})

0

Solution steps

x \to 0 + lim (x^{9} (ln (x))^{2})

Rewrite for L'Hopital

= x \to 0 + lim (\frac{( ln ( x ) ) ^{2}}{\frac{1}{x ^{9}}})

Apply L'Hopital's Rule

: x \to 0 + lim (\frac{\frac{2 l n ( x )}{x}}{- \frac{9}{x ^{10}}})

= x \to 0 + lim (\frac{\frac{2 l n ( x )}{x}}{- \frac{9}{x ^{10}}})

Simplify

\frac{\frac{2 l n ( x )}{x}}{- \frac{9}{x ^{10}}} : - \frac{2}{9} x^{9} ln (x)

= x \to 0 + lim (- \frac{2}{9} x^{9} ln (x))

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

= - \frac{2}{9} \cdot x \to 0 + lim (x^{9} ln (x))

Rewrite for L'Hopital

= - \frac{2}{9} \cdot x \to 0 + lim (\frac{ln ( x )}{\frac{1}{x ^{9}}})

Apply L'Hopital's Rule

: x \to 0 + lim (\frac{\frac{1}{x}}{- \frac{9}{x ^{10}}})

= - \frac{2}{9} \cdot x \to 0 + lim (\frac{\frac{1}{x}}{- \frac{9}{x ^{10}}})

Simplify

\frac{\frac{1}{x}}{- \frac{9}{x ^{10}}} : - \frac{x ^{9}}{9}

= - \frac{2}{9} \cdot x \to 0 + lim (- \frac{x ^{9}}{9})

Plug in the value

x = 0

= - \frac{2}{9} (- \frac{0 ^{9}}{9})

Simplify

- \frac{2}{9} (- \frac{0 ^{9}}{9}) : 0

= 0

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What is the limit as x approaches 0+of x^9(ln(x))^2 ?
The limit as x approaches 0+of x^9(ln(x))^2 is 0