Answer
$\underset{x\to -\frac{\pi }{2}}{\mathop{\lim }}\,\sin x=-1$, since, as x approaches $-\frac{\pi }{2}$ the value of $ f\left( x \right)$ gets closer to the y-coordinate of $-1$.
Work Step by Step
Consider the provided limit, $\underset{x\to -\frac{\pi }{2}}{\mathop{\lim }}\,\sin x $.
To find $\underset{x\to -\frac{\pi }{2}}{\mathop{\lim }}\,\sin x $ examine the portion of the graph near $ x=-\frac{\pi }{2}$.
As x approaches $-\frac{\pi }{2}$ the value of $ f\left( x \right)$ gets closer to the y-coordinate of $-1$. This point $\left( -\frac{\pi }{2},-1 \right)$ is shown by the solid dot in the above graph.
The point $\left( -\frac{\pi }{2},-1 \right)$ has a y-coordinate of $-1$.
Thus, $\underset{x\to -\frac{\pi }{2}}{\mathop{\lim }}\,\sin x=-1$.
Hence, the value of $\underset{x\to -\frac{\pi }{2}}{\mathop{\lim }}\,\sin x $ is $-1$ .
