Why do parabolas' arms eventually become parallel? Here is what this site states

All parabolas are the same shape, no matter how big they are. Although they are infinite, meaning that the arms will never close up, the arms will eventually become parallel.

Now, I have an argument against it. Let $f(x) = ax^2 + bx + c$ be a quadratic polynomial with $a , b$ and $c$ being real numbers and $a \ne 0$. So, its graph will give us a parabola ($\because$ graph of a quadratic polynomial is a parabola). 
Now,
$$\dfrac{d (ax^2 + bx + c)}{dx}$$$$ = 2ax + b$$
i.e. the slope of a quadratic polynomial is given by $g(x) = 2ax + b$. Now, differentiating the equation for slope of the quadratic
$$\dfrac {d (2ax + b)}{d x}$$
$$ = 2a$$
So, if $a \gt 0$ then the slope of $g(x)$ will be increasing. This means that the slope of $f(x)$ will also be increasing. Similarly, if $a \lt 0$ then the slope of $f(x)$ will be decreasing. 
This means that for all $x$ the slope of $f(x)$ will be different. So, this contradicts the fact (according to me) that the arms of a parabola will eventually be parallel. Where am I going wrong?

The precise sense in which the statement is correct is that when viewed from a great distance above the plane of the parabola, thought of as a dynamic process of "zooming out to infinity", two things will happen (not necessarily at the same speed, but both processes will become visible given enough distance of the viewer from the plane).


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*The two arms start to look flat and linear, like two straight lines with the parabola's axis of symmetry bisecting the angle between them.

*The angle between those two lines tends to 0, until the arm "lines" look indistinguishable from the axis of symmetry.
If there are several parabolas in the plane with parallel axes of symmetry then their "arms" will all appear to collapse into one line when seen from a great enough perpendicular distance to the plane.
