The paradox, as it was presented to us in our Real Analysis class, was to take a right isoceles triangle and lay it down on its hypotenuse. Then, take the vertex at the right angle, and fold that down to where it touched the hypotenuse, thus making two right isoceles triangles. Keep doing this infinitely many times, and you can establish that 2 = SQRT(2).
I’m no mathmatician but how about navigating long distances at sea before global positioning?
Surely spherical trigonometry comes into play?
This is the second time somebody mentioned indeterminate forms as if they are final. L’Hopital’s rule is a great way to take the limit of most indeterminate forms.
In your case, though, we needn’t use L’Hopital’s rule, as it is obvious that the limit of f(z), as z -> infinity, is d/sqrt(2). (The z’s cancel, there’s nothing indeterminate about it.)
I agree with your conclusion but the math is getting slightly over my head (I learned L’Hopital’s rule but it was 32 years ago. :o ). But we seem to be coming full circle in the thread. **Eonwe ** is claiming that in this case,
limit of f(z) as z -> infinity
is different than
f(z) when z = infinity
Is that possible?
Better to write it as f[sub]n/sub, and consider the limit as n goes to infinity. This sort of wonky limiting behavior is usually due to a sequence of functions converging but not converging uniformly. I don’t know if that’s what’s going on here, but it’s a reasonable first guess.