Answer
a) The limit provided in part(a) is not correct.
b) The limit provided in part(b) is not correct.
c) The limit provided in part(c) is not correct.
d) The limit provided in part(d) is correct.
Work Step by Step
L'Hospital's rule states that $\lim\limits_{x \to \infty} f(x)=\lim\limits_{x \to \infty} \dfrac{A'(x)}{B'(x)}$
a) Here, $\lim\limits_{x \to 0^{+}} x \ln x\ne (0) (-\infty )=0$
Incorrect.
b) Here, $\lim\limits_{x \to 0^{+}} x \ln x\ne (0) (-\infty )=-\infty$
Incorrect.
c) Here,$\lim\limits_{x \to 0^{+}} x \ln x =\lim\limits_{x \to 0^{+}} \dfrac{\ln x}{1/x}\ne \dfrac{-\infty}{\infty}=-1$
Incorrect.
d) Here, $\lim\limits_{x \to 0^{+}} x \ln x =\lim\limits_{x \to 0^{+}} \dfrac{\ln x}{1/x}=\lim\limits_{x \to 0^{+}} (-x) =0$
Correct.