I would like to compute the equivalent function when $k\rightarrow 0$ of $$\frac{1}{k}\,\left(\,Z(\frac{u}{k},k) - dn(\frac{u}{k},k)\,\right)$$ where $k$ is the modulus of the Jacobi elliptic function $dn$ and $Z$ the elliptic functions of second species Zeta and $u$ a real variable.
Numerically the limit exists but I can't find any reference for help. I look into the Lawden's book, I find this kind of relation: $dn (\frac{u}{k},k)= cn(u,\frac{1}{k})$ but the schmilblick is still at the same place.
Note that: $dn^2(u/k,k) = 1-k^2sn^2(u/k,k)$ and $sn(u/k,k)$ bounded so $1/k\,dn^2(u/k,k)$ behaves like $1/k$ when $k\rightarrow 0$.