How quickly does the distance between the parabola y=x^2 and y=x^2+1 change as you increase x?

Question

asnaseer · Answer

by "distance" do you mean the perpendicular distance between the tangent lines at a given value for $x$?

michele_laino · Answer

I think that distace is constant and equal to 1

michele_laino · Answer

oops...distance...

asnaseer · Answer

it is constant if you interpret the distance as the vertical distance between the two curves at any given $x$

michele_laino · Answer

I used this formula:
$$d=\sup|(f(x)-g(x)|$$

asnaseer · Answer

or does @Kainui mean the "shortest" distance between the two curves as we increase $x$?

michele_laino · Answer

is the max rate of increasing =1?

michele_laino · Answer

oops not 1, the formula is:
$$\lim _{x \rightarrow +\infty}\frac{ 4 }{ \sqrt{4+(1/x ^{2})} }=$$

michele_laino · Answer

$$=2$$

asnaseer · Answer

I think we first need Kainui to clarify what he/she means by "distance" here

kainui · Answer

By distance I mean shortest length between one parabola to the other.