Answer
$e^3$
Work Step by Step
Our aim is to evaluate the limit for $\lim\limits_{x \to 0^{+}} (\sin x)^{3/\ln x}$
Let us consider that $y=\lim\limits_{x \to 0^{+}} (\sin x)^{3/\ln x} \implies \ln y=\lim\limits_{x \to 0^{+}}\dfrac{3}{\ln x} \times \ln (\sin x)$
Apply L-Hospital's rule which can be defined as: $\lim\limits_{x \to a} \dfrac{P(x) }{Q(x)}=\lim\limits_{x \to a}\dfrac{P'(x)}{Q'(x)}$
$ \ln y=\lim\limits_{x \to 0^{+}}\dfrac{3 x \cos x}{\sin x}\\=3 \lim\limits_{x \to 0^{+}} (\cos x) \times \lim\limits_{x \to 0^{+}} \dfrac{x}{\sin x}\\=(3)(1)(1) \\=3$
Thus, we have: $\ln y=3\implies y=e^3$
