Question

$\underset{x\to 0+}{lim}\left(x^n\ln x\right),n>0$

• A. does not exist
• B. exists and is zero
• C. exists and is $1$
• D. exists and is $e^{-1}$

Answer 1

The correct option is B exists and is zero

Let $h>0$
$\underset{h\to 0}{lim}{h}^n\ln h=\underset{h\to 0}{lim}\left(\frac{\ln h}{\frac{1}{h^n}}\right)$
$=\underset{h\to 0}{lim}\left(\frac{\frac{1}{h}}{\frac{-n}{h^{n+1}}}\right)$ (using L'hospital rule)
$=\underset{h\to 0}{lim}\left(\frac{h^n}{-n}\right)=0$