## Calculus 10th Edition

$f(x)$ is continuous over the interval $(-\infty, \infty).$
Using Theorem $1.11:$ $f(x)=\dfrac{g(x)}{h(x)}\to g(x)=x$ and $h(x)=x^2+x+2.$ $f(x)$ is continuous as long as both $g(x)$ and $h(x)$ are continuous and $h(x)\ne0.$ $g(x)$ is continuous over the interval $(-\infty, \infty)$ and $h(x)$ is continuous over the interval $(-\infty, \infty)$; furthermore, $h(x)$ is never equal to zero since the determinant of the quadratic, $\Delta=(1)^2-4(1)(2)=-7\to$ no real roots. All this shows that $f(x)$ is continuous over the interval $(-\infty, \infty).$