Q.Find the range of the following function: $f(x)=\dfrac{x}{9+x^2}$.

A rational function $f(x)=\dfrac{p(x)}{q(x)}$ is a ratio of two polynomials, defined wherever the denominator $q(x)\ne0$ — so its domain is always $R$ minus the (finite) set of zeroes of $q(x)$. The simplest example, $f(x)=\dfrac1x$, introduces two key ideas: as $x\to0$, $f(x)\to\pm\infty$, so the line $x=0$ is a vertical asymptote (a line the curve approaches but never touches, right where the denominator vanishes); and as $x\to\pm\infty$, $f(x)\to0$, so the line $y=0$ is a horizontal asymptote (the value the function settles toward for very large or very negative $x$). …