Question

\(\lim _{x \rightarrow 0+}\left(x^{n} \ln x\right), n>0\)
(A) does not exist
(B) exists and is zero
(C) exists and is 1
(D) exists and is \(\mathrm{e}^{-1}\)

Answer 1

Ans: (B)
Hint \(: \operatorname{Lt}_{x \rightarrow 0^{+}} \frac{\ln x}{\frac{1}{x^{n}}}\left(\frac{\infty}{\infty}\right)\). Applying LH rule
$$
\Rightarrow \operatorname{Lt}_{x \rightarrow 0^{+}} \frac{\frac{1}{x}}{\frac{-n}{x^{n+1}}}=0
$$