What is the limit as x approaches 0+of x(ln(x))^6 ?

The limit as x approaches 0+of x(ln(x))^6 is 0

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limit as x approaches 0+of x(ln(x))^6

x \to 0 + lim (x (ln (x))^{6})

0

Solution steps

x \to 0 + lim (x (ln (x))^{6})

Rewrite for L'Hopital

= x \to 0 + lim (\frac{( ln ( x ) ) ^{6}}{\frac{1}{x}})

Apply L'Hopital's Rule

: x \to 0 + lim (\frac{\frac{6 l n ^{5} ( x )}{x}}{- \frac{1}{x ^{2}}})

= x \to 0 + lim (\frac{\frac{6 l n ^{5} ( x )}{x}}{- \frac{1}{x ^{2}}})

Simplify

\frac{\frac{6 l n ^{5} ( x )}{x}}{- \frac{1}{x ^{2}}} : - 6 x ln^{5} (x)

= x \to 0 + lim (- 6 x ln^{5} (x))

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

= - 6 \cdot x \to 0 + lim (x ln^{5} (x))

Rewrite for L'Hopital

= - 6 \cdot x \to 0 + lim (\frac{ln ^{5} ( x )}{\frac{1}{x}})

Apply L'Hopital's Rule

: x \to 0 + lim (\frac{\frac{5 l n ^{4} ( x )}{x}}{- \frac{1}{x ^{2}}})

= - 6 \cdot x \to 0 + lim (\frac{\frac{5 l n ^{4} ( x )}{x}}{- \frac{1}{x ^{2}}})

Simplify

\frac{\frac{5 l n ^{4} ( x )}{x}}{- \frac{1}{x ^{2}}} : - 5 x ln^{4} (x)

= - 6 \cdot x \to 0 + lim (- 5 x ln^{4} (x))

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

= - 6 (- 5 \cdot x \to 0 + lim (x ln^{4} (x)))

Rewrite for L'Hopital

= - 6 (- 5 \cdot x \to 0 + lim (\frac{ln ^{4} ( x )}{\frac{1}{x}}))

Apply L'Hopital's Rule

: x \to 0 + lim (\frac{\frac{4 l n ^{3} ( x )}{x}}{- \frac{1}{x ^{2}}})

= - 6 (- 5 \cdot x \to 0 + lim (\frac{\frac{4 l n ^{3} ( x )}{x}}{- \frac{1}{x ^{2}}}))

Simplify

\frac{\frac{4 l n ^{3} ( x )}{x}}{- \frac{1}{x ^{2}}} : - 4 x ln^{3} (x)

= - 6 (- 5 \cdot x \to 0 + lim (- 4 x ln^{3} (x)))

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

= - 6 (- 5 (- 4 \cdot x \to 0 + lim (x ln^{3} (x))))

Rewrite for L'Hopital

= - 6 (- 5 (- 4 \cdot x \to 0 + lim (\frac{ln ^{3} ( x )}{\frac{1}{x}})))

Apply L'Hopital's Rule

: x \to 0 + lim (\frac{\frac{3 l n ^{2} ( x )}{x}}{- \frac{1}{x ^{2}}})

= - 6 (- 5 (- 4 \cdot x \to 0 + lim (\frac{\frac{3 l n ^{2} ( x )}{x}}{- \frac{1}{x ^{2}}})))

Simplify

\frac{\frac{3 l n ^{2} ( x )}{x}}{- \frac{1}{x ^{2}}} : - 3 x ln^{2} (x)

= - 6 (- 5 (- 4 \cdot x \to 0 + lim (- 3 x ln^{2} (x))))

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

= - 6 (- 5 (- 4 (- 3 \cdot x \to 0 + lim (x ln^{2} (x)))))

Rewrite for L'Hopital

= - 6 (- 5 (- 4 (- 3 \cdot x \to 0 + lim (\frac{ln ^{2} ( x )}{\frac{1}{x}}))))

Apply L'Hopital's Rule

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

= - 6 (- 5 (- 4 (- 3 \cdot 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)

= - 6 (- 5 (- 4 (- 3 \cdot x \to 0 + lim (- 2 x ln (x)))))

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

= - 6 (- 5 (- 4 (- 3 (- 2 \cdot x \to 0 + lim (x ln (x))))))

Rewrite for L'Hopital

= - 6 (- 5 (- 4 (- 3 (- 2 \cdot x \to 0 + lim (\frac{ln ( x )}{\frac{1}{x}})))))

Apply L'Hopital's Rule

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

= - 6 (- 5 (- 4 (- 3 (- 2 \cdot x \to 0 + lim (\frac{\frac{1}{x}}{- \frac{1}{x ^{2}}})))))

Simplify

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

= - 6 (- 5 (- 4 (- 3 (- 2 \cdot x \to 0 + lim (- x)))))

Plug in the value

x = 0

= - 6 (- 5 (- 4 (- 3 (- 2 (- 0)))))

Simplify

= 0

x \to 5 lim (\frac{x ^{2} - 2}{x - 5})

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

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

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

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

What is the limit as x approaches 0+of x(ln(x))^6 ?
The limit as x approaches 0+of x(ln(x))^6 is 0