Answer
The limit does not exist.
Work Step by Step
Applying the limit laws for a sum and a constant, if the limit L exists, then
$L= 2-\displaystyle \lim_{\theta\rightarrow 0}\cot\theta$
You can 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 through the 4th quadrant, where $\cot\theta$ is negative, and
$\displaystyle \lim_{\theta\rightarrow 0^{-}}\cot\theta=-\infty$
When $\theta\rightarrow 0^{+}$, it approaches 0 through the 1st quadrant, where $\cot\theta$ is positive, and
$\displaystyle \lim_{\theta\rightarrow 0^{+}}\cot\theta=+\infty$
Since the one-sided limits differ, we say that the limit does not exist.
