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Let $f(x)=\frac{x}{5 \sin (x+1)}$.$\newline$Select the correct description of the one-sided limits of $f$ at $x=-1$.$\newline$Choose $1$ answer:$\newline$(A)$\newline$$$\begin{array}{l} \lim _{x \rightarrow-1^{+}} f(x)=+\infty \text { and } \\ \lim _{x \rightarrow-1^{-}} f(x)=+\infty \end{array}$$$\newline$(B)$\newline$$$\begin{array}{l} \lim _{x \rightarrow-1^{+}} f(x)=+\infty \text { and } \\ \lim _{x \rightarrow-1^{-}} f(x)=-\infty \end{array}$$$\newline$(C)$\newline$$$\begin{array}{l} \lim _{x \rightarrow-1^{+}} f(x)=-\infty \text { and } \\ \lim _{x \rightarrow-1^{-}} f(x)=+\infty \end{array}$$$\newline$(D)$\newline$$$\begin{array}{l} \lim _{x \rightarrow-1^{+}} f(x)=-\infty \text { and } \\ \lim _{x \rightarrow-1^{-}} f(x)=-\infty \end{array}$$