As part of a calculation in particle physics, I have encountered the evaluation $f(\vec{k},0)$ of the following function of two vector variables:
$$f(\vec{k}_1,\vec{k}_2)=\dfrac{\vec{k}_1\cdot\vec{k}_2}{k_1^2k_2^2}$$
This seems to produce a $0/0$ indetermination. I must admit I am completely lost. I am familiar with the usual techniques for calculating 2D and 3D limits but as far as I can tell this looks like the result will be $\pm\infty$, and the limit therefore does not exist. Is this correct? I reached this conclusion by trying to compute the limit along the $z$ axis and getting, by plugging $k_{2x}=k_{2y}=0$, that:
$$\lim_{(k_{2x},k_{2y},k_{2z})\to(0,0,0)}\dfrac{k_{1x}k_{2x}+k_{1y}k_{2y}+k_{1z}k_{2z}}{(k_{1x}^2+k_{1y}^2+k_{1z}^2)(k_{2x}^2+k_{2y}^2+k_{2z}^2)}=$$ $$=\lim_{k_{2z}\to 0}\dfrac{k_{1z}}{(k_{1x}^2+k_{1y}^2+k_{1z}^2)k_{2z}}=\pm\infty$$
Any suggestions, ideas or corrections would be greatly appreciated!