Answer
$$\lim_{\theta\to0^-}(1+\csc\theta)=-\infty$$
Work Step by Step
$$A=\lim_{\theta\to0^-}(1+\csc\theta)=1+\lim_{\theta\to0^-}\csc\theta=1+\lim_{\theta\to0^-}\frac{1}{\sin\theta}$$
As $\theta\to0^-$, $\sin x$ approaches $\sin0=0$ from the left, where $\sin\theta\lt0$.
($\theta\to0^-$ are values of $\theta$ like $-\pi/6, -\pi/4$, etc. and all these values give $\sin \theta$ negative values)
Therefore, $1/\sin \theta$ will approach $-\infty$. In other words,
$$A=1+(-\infty)=-\infty$$
