For a (infinitely, if necessary) differentiable curve $$ A(t) = (x(t), y(t)) $$ which diverges at $t_0 \in [-\infty,\infty] $, that is $$ \lim_{t \to t_0 } | x(t)^2 + y(t)^2 | =\infty $$ if there is a line $l: ax+by+c=0$ such that $$ \lim_{t \to t_0 } \frac{ |ax(t) +by(t)+ c | }{\sqrt{a^2+b^2} } = 0 $$ then this line $l$ is called the asymptotic line of curve $A(t)$ as $ t \to t_0$
I want to verify that the tangent vector of $A(t)$ $$A'(t) = (x'(t) , y' (t)) $$ tends to become parallel to line $l$ as $t \to t_0 $
It seems intuitively obvious and not difficult to prove at first glance but I have no idea how to approach..
Thanks in advance.
