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
(a )Here, $\lim\limits_{x \to 0^{+}} x \ln x\ne (0) (-\infty )=0$
We cannot use L-Hospital's rule because we do not get any indeterminate form of limit.
(b) Here, $\lim\limits_{x \to 0^{+}} x \ln x\ne (0) (-\infty )=-\infty$
We cannot use L-Hospital's rule because we do not get any indeterminate form of limit.
(c) Here,$\lim\limits_{x \to 0^{+}} x \ln x =\lim\limits_{x \to 0^{+}} \dfrac{\ln x}{1/x}\ne \dfrac{-\infty}{\infty}=-1$
We cannot use L-Hospital's rule because we do not get any indeterminate form of limit.
(d) Here, $\lim\limits_{x \to 0^{+}} x \ln x =\lim\limits_{x \to 0^{+}} \dfrac{\ln x}{1/x}=\lim\limits_{x \to 0^{+}} (-x) =0$
