Question

How do you prove that the function `f(x) = sqrt(x)` is continuous at 0 to infinity?

Answer 1

You have to prove that:

`lim_(Deltax->0) f(x+Deltax) - f(x) = 0`

Explanation:

We have:

`sqrt(x+Deltax) -sqrt(x) = (sqrt(x+Deltax) -sqrt(x))(sqrt(x+Deltax) +sqrt(x))/(sqrt(x+Deltax) +sqrt(x)) = ((x+Deltax) -x)/(sqrt(x+Deltax) +sqrt(x)) = (Deltax)/(sqrt(x+Deltax) +sqrt(x))`

So that:

`lim_(Deltax->0) sqrt(x+Deltax) -sqrt(x) = lim_(Deltax->0) (Deltax)/(sqrt(x+Deltax) +sqrt(x)) = 0`

And the continuity is proved.