What is the limit as x approaches 0+of x^2log_{e}(x) ?

The limit as x approaches 0+of x^2log_{e}(x) is 0

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limit as x approaches 0+of x^2log_{e}(x)

x \to 0 + lim (x^{2} lo g_{e} (x))

0

Solution steps

x \to 0 + lim (x^{2} lo g_{e} (x))

Rewrite for L'Hopital

= x \to 0 + lim (\frac{lo g _{e} ( x )}{\frac{1}{x ^{2}}})

Apply L'Hopital's Rule

= 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}

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

Plug in the value

x = 0

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

Simplify

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

= 0

What is the limit as x approaches 0+of x^2log_{e}(x) ?
The limit as x approaches 0+of x^2log_{e}(x) is 0