Study of y = x/(x²+1)

Text

Study $y=\dfrac{x}{x^{2}+1}$: domain, symmetry, sign, asymptotes, maxima and minima, inflections.

Solution

Domain and symmetry. Domain $\mathbb{R}$ (denominator always positive); the function is odd. Sign: $y>0$ for $x>0$, $y<0$ for $x<0$, zero at $x=0$. Asymptotes. $\displaystyle\lim_{x\to\pm\infty}\frac{x}{x^{2}+1}=0$: horizontal asymptote $y=0$. Extrema. $y'=\dfrac{1-x^{2}}{(x^{2}+1)^{2}}$, vanishing at $x=\pm1$. At $x=1$ maximum $y=\tfrac12$; at $x=-1$ minimum $y=-\tfrac12$. Inflections. $y''=\dfrac{-2x\,(3-x^{2})}{(x^{2}+1)^{3}}$ vanishes at $x=0$ and $x=\pm\sqrt3$: three inflections. $\boxed{\text{max }\left(1,\tfrac12\right);\quad \text{min }\left(-1,-\tfrac12\right);\quad \text{asymptote } y=0}$