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?