I understand how to create a horizontal phase line of a function...

$\displaystyle \frac{dx}{dt}=5x^2(x+5)(x-3)^3$

--->-----(-5)-----<------(0)------<------(3)----->----

...but I don't really understand what it means. What does the horizontal line represent and what do the arrows represent?

And what procedure would I use for answering the question:

The solution of the above equation with $\displaystyle x(0)=-4$ is defined on the interval $\displaystyle a<t<b $, find

$\displaystyle lim_{t\to a+} x(t)= $

$\displaystyle lim_{t\to b-} x(t)= $

Is there a way to find the limit analytically? I'm guessing not since we're using the phase line to do it.