Answer
$-\infty$
Work Step by Step
Applying the limit laws for a sum and a constant, if the limit L exists, then
$L=1+\displaystyle \lim_{\theta\rightarrow 0^{-}}\frac{1}{\sin\theta}$
You now might sketch a unit circle. 0 radians is at (1,0)$;$ the positive direction is counterclockwise; the negative direction is clockwise.
When $\theta\rightarrow 0^{-}$, it approaches 0 from the NEGATIVE side.
$\theta$ is in quadrant IV (clockwise from 0), where sine is negative.
Thus, $\displaystyle \csc x=\frac{1}{\sin\theta}\rightarrow-\infty$
So, L does not exist, but we can say
$\displaystyle \lim_{\theta\rightarrow 0^{-}}(1+\csc x)=-\infty$
