Osculating circle of plane curve at a point where the curvature is 0

Question

The problem is

Find an equation of the osculating circle of $f(x) = \arctan{x}$ at the origin.

So I calculate it's curvature and got

$\kappa(x) = \frac{2(x^3 + x)}{(x^4 + 2x^2 + 2)^{3/2}}$.

Thus $\kappa(0) = 0$. As I know the radius of the osculating circle at a point $x_0$ is $\frac{1}{\kappa(x_0)}$, which means the radius of the circle in the question is not defined.

What should I do?

Answer 1

If you consider the osculating circle as it approaches the origin from either side, what you will notice is the circle is on opposite sides of the curve. As it approaches the origin, it gets bigger and bigger and "jumps" to the other side of the curve at the origin.

This is the reason the osculating circle is not defined at the origin, so you could conclude that is the correct answer.