A multivariable limit must agree along every route to the target.
The first function has line-path limits k/(1+k²), so zero and diagonal approaches disagree. The second function tends to zero along every fixed straight line, but along y=kx² it equals k/(1+k²). Thus even checking every straight line can miss a failed multivariable limit.
A collection of agreeing paths is evidence, not a proof of a full multivariable limit.