Study of the rational function y = x over (x squared plus 1)

Text

Study the function $y=\dfrac{x}{x^{2}+1}$: domain, symmetry, asymptotes, monotonicity and extrema.

Solution

Domain $\mathbb{R}$; odd function. Intercepts: only the origin $(0,0)$. Horizontal asymptote: $\displaystyle\lim_{x\to\pm\infty}\dfrac{x}{x^{2}+1}=0$, so $y=0$. First derivative: $y'=\dfrac{1-x^{2}}{(x^{2}+1)^{2}}=0\Rightarrow x=\pm1$. $y'>0$ on $(-1,1)$: max $(1,\tfrac12)$, min $(-1,-\tfrac12)$. $\boxed{\text{max }(1,\tfrac12);\ \text{min }(-1,-\tfrac12);\ \text{asymptote }y=0}$