Question

Evaluate: $\underset{x\to 0^{+}}{lim}{x}^3\log x$

Answer 1

$\underset{x\to 0^{+}}{lim}{x}^3{\log }_{e}x$

$=\underset{x\to 0^{+}}{lim}\left(\frac{{\log }_{e}x}{\frac{1}{x^3}}\right)\left(\frac{\infty }{\infty }\text{form}\right)$

Applying L Hospital's rule, we have

$=\underset{x\to 0^{+}}{lim}\left(\frac{\frac{1}{x}}{\frac{-3}{x^4}}\right)=\underset{x\to 0^{+}}{lim}\left(\frac{x^4}{-3x}\right)$

$=\underset{x\to 0^{+}}{lim}\left(\frac{-x^3}{3}\right)=0$