Sketch x/(1+x^2)

To sketch the curve, first note that $f(x)=-f(x)$, so the curve is anti-symmetrical about the origin. Also, find that $f(0)=0$, and $\lim_{x\to\infty}f(x)=0$.

The turning point can be found by setting $\frac{x}{1+x^2}=c$, rearranging to $$ c x^2 - x + c = 0 $$ This equation has exactly one solution when the line $y=c$ passes through the turning point, and so the discriminant must be equal to 0. Therefore $$ \begin{aligned} 1 - 4c^2 &= 0 \\ c &= \pm \frac{1}{2} \\ x &= \pm 1 \end{aligned}$$