I was asked to prove that the derivative at x=0 of f(x)=x^2/3 does not exist. I know that it doesn't because the function is a corner at x=0, but I'm not sure how to start. I know the definition of derivative and I believe I need to do this by contradiction using the definition and cases, but what do I do now?
I have only proved that a limit is a real number. I can see that the limit as h-->0 from the left is negative infinity and as h-->0 from the right the limit is positive infinity. Since there isn't a unique limit then the limit DNE. But, I don't know how to show a limit is infinity positive or negative. Help?
If the limit does not exist
i.e. Limit from the postive =/= limit from the negative, then the function isn't continuous.
For a funtion to be differentiable at a point, it needs to be continuous at that same point, thus the limit needs to exist at that point.
You've shown that it isn't continuous at that point, then by definition it cannot be differentiate at that point.
Okay, I figured this out. So when I show there is a delta so that 0<x<delta, and let delta equal 1/N^3, then N</= 1/3rooth, that means that the limit is positive infinity?
What about the left sided limit that goes to negative infinity? How do I do that?
Isn't that bit instrumental in showing that at 0 there are two different limits and therefor the limit doesn't exist at zero? And since the limit doesn't exist the original function has no derivative at x=0?
Thanks for the help Plato.