Answer
$$
\lim _{x \rightarrow 0^{+}} \ln (\sinh x)=-\infty
$$
Work Step by Step
$$
\lim _{x \rightarrow 0^{+}} \ln (\sinh x)
$$
Let
$$
t=\sinh x, \text { As } x \rightarrow 0^{+}, t \rightarrow 0^{+},
$$
So,
$$
\lim _{x \rightarrow 0^{+}} \ln (\sinh x)=\lim _{t \rightarrow 0^{+}} \ln t=-\infty
$$
