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

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

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

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

0

Solution steps

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

ln (x^{2}) = 2 ln (x)

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

Rewrite for L'Hopital

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

Apply L'Hopital's Rule

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

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

Simplify

\frac{\frac{1}{x}}{- \frac{2}{x ^{3}}} : - \frac{x ^{2}}{2}

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

Plug in the value

x = 0

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

Simplify

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

= 0

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

x \to 0 lim (tan (x) (ln (x)))

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

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

x \to 0 + lim (5 x^{x})

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