Question

Evaluate the following limit​

Answer 1

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$\lim_ {x \to0^{ + } }( {x}^{2} log(x) )$

$= \lim_ {x \to0^{ + } }( \frac{1}{ \frac{1}{ {x}^{2} } } log(x) )$

$= \lim_ {x \to0^{ + } }( \frac{ log(x) }{ \frac{1}{ {x}^{2} } } )$

$= \lim_ {x \to0^{ + } }( \frac{ \frac{d}{dx} log(x) }{ \frac{1}{ \frac{d}{dx} \frac{1}{ {x}^{2} } } } )$

$= \lim_ {x \to0^{ + } }( \frac{ \frac{1}{ ln(10) x} }{ - \frac{2}{ {x}^{3} } })$

$= \lim_ {x \to0^{ + } }( - \frac{1}{x \: ln(10) } \times \frac{ {x}^{3} }{2} )$

$= \lim_ {x \to0^{ + } }( - \frac{ {x}^{2} }{2 ln(10) } )$

$= - \frac{ {0}^{2} }{2 \times 1}$

$= 0$

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• Answer is 0.