in do carmo's differential geometry of curves and surfaces he has the following problem:
let be a continuous and simple (one to one) curve. we say has a weak tangent at if the line determined by and has a limit position as h approaches 0. we say has a strong tangent at if the line determined by and has a limit position as h, k approach 0. show that has a weak tangent but no strong tangent at t = 0.
so for the case of a weak tangent, i have that and . so
in the case for a strong tangent i have and . then
but it seems to me that both have limit positions so both the weak and strong tangent exist at t = 0? i am not too clear on the definition he gives in the problem of weak and strong tangent. what does he exactly mean by "limiting position". if he means that the line will obtain a fixed "direction" then wouldn't neither a weak nor a strong tangent exist in this case since both lines in this case just go to (0, 0) so actually there isn't even a line at all, but just a point or the zero vector?