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

1 Answer
Jan 19, 2017

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.