Answer
continuous on $(-\infty, -1)\cup(-1, \infty)$.
Work Step by Step
Let f(x)=$\sin$x.
It is continuous everywhere, by theorem 7
(polynomials, rational functions, root functions, trigonometric functions are continuous on their domains)
Let $g(x)=x+1$, , a polynomial,
continuous everywhere, by theorem 7.
$h(x)=\displaystyle \frac{f(x)}{g(x)}$ has domain: $g(x)\neq 0$
$x\neq-1$
Domain= $ (-\infty, -1)\cup(-1, \infty)$.
It is continuous on its domain by Th.4.5
(If $f$ and $g$ are continuous, then $\displaystyle \frac{f}{g}$ is continuous on its domain.
