What is the limit as x approaches 0+of x^3(ln(x))^2 ?

The limit as x approaches 0+of x^3(ln(x))^2 is 0

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limit as x approaches 0+of x^3(ln(x))^2

x \to 0 + lim (x^{3} (ln (x))^{2})

0

Solution steps

x \to 0 + lim (x^{3} (ln (x))^{2})

Rewrite for L'Hopital

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

Apply L'Hopital's Rule

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

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

Simplify

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

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

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

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

Rewrite for L'Hopital

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

Apply L'Hopital's Rule

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

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

Simplify

\frac{\frac{1}{x}}{- \frac{3}{x ^{4}}} : - \frac{x ^{3}}{3}

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

Plug in the value

x = 0

= - \frac{2}{3} (- \frac{0 ^{3}}{3})

Simplify

- \frac{2}{3} (- \frac{0 ^{3}}{3}) : 0

= 0

y = x (1 + x^{2})^{10}

x \to 0 - lim (\frac{x}{x})

\int (3 y - 4 y^{3})^{3} d y

y = ln (\frac{x}{2}), at x = 4

\int \frac{t ^{2}}{t + 1} d t

What is the limit as x approaches 0+of x^3(ln(x))^2 ?
The limit as x approaches 0+of x^3(ln(x))^2 is 0