A function with strange behavior near the boundary of domain

Question

During some calculations, I came up with a very weird function. It reads

$$f(k)=\frac{8(1-k^2)+k^4+4(k^2-2)\sqrt{1-k^2}}{k^4\sqrt{1-k^2}}.$$

Fo $f$ to be real, we have to choose $k\in [-1,0[ \ \cup \ ]0,1]$. Depending on the region we choose to represent this function, the several plots I get are really different from each other and get really strange. For example, for $k\in [0.01,0.99]$, and using $N=100$ sampling points, I get

Above, the function seems to have a very normal behavior. However, when I evaluate it near $k=0$, the functions seems highly discontinuous. For $k \in [1,2]\times10^{-4}$, I get

Another example is $k\in [1,10]\times10^{-9}$, for which I get the weirdest plot of all,

Can someone elucidate me on this behavior? What kind of functions is this, and why does this happen? By the way, I am using Python but I don't think this is a numerical problem.

Moreover, I am interested in obtaining an approximation for $f$ near $k=0$. These plots seem to indicate that such a function holds no approximation near the origin. However, the Taylor expansion function of Mathematica gives me this

Is this expression valid near $k=0$?

Answer 1

Yes, it is a numeric problem. You have$$f(k)=\frac{8\sqrt{1-k^2}+4(k^2-2)}{k^4}+\frac1{\sqrt{1-k^2}}$$when $k\in(-1,1)\setminus\{0\}$. The $\frac1{\sqrt{1-k^2}}$ part causes no problem near $0$. And, again near $0$, you have$$8\sqrt{1-k^2}+4(k^2-2)=-k^4-\frac12k^6+\cdots,$$and therefore$$\frac{8\sqrt{1-k^2}+4(k^2-2)}{k^4}=-1-\frac12k^2+\cdots$$So, if you decide that $f(0)=0$, you get a continuous functions from $(-1,1)$ into $\Bbb R$.