I have a problem with ODE.
Let's suppose I have a first order ODE, such
$$y' = f(x, y)$$
Let's suppose $f$ is satisfy the required hypothesis for uniqueness and existance in $\mathbb{R}$
Let's also suppose that $f(x, y) \ge 0$, so that $y$ is always increasing.
Let $y(0) = y_0 < 0$
Now, I want to estabilish if $y$ has an asymptote in, let's say, $1$.
I suppose that $\lim_{x \to \infty} y = 1$ and find out that $\lim_{x \to \infty} y' = 0$
But this condition is only necessary.
How can I be sure that there is an asymptote in $1$ or not? What should I check about $f(x, y)$?
On the other hand, if I suppose
$\lim_{x \to \infty} y = +\infty$, and I find that $\lim_{x \to \infty} y' = 0$, I cannot conclude that it has an asymptote right? I am basically stuck.
Thank in advance for your help.