https://www.cut-the-knot.org/wiki-math/index.php?n=Calculus.DerivativesWithoutLimits

..and thought it was rather interesting. Then I wondered if one could show that a function is not differentiable without limits. Well here it goes:

Problem: Find the slope of the tangent line to the graph of y = x^(2/3) at the point (0,0).

Solution: The line must pass through the y-axis at 0, so let y = mx be the equation of the line. Then we have:

x^(2/3) = mx

x^2 = (mx)^3

m = (1/x)^(1/3) which is undefined at x = 0

Therefore the slope of the tangent line to the graph of y = x^(2/3) at (0,0) has a slope that is undefined, and is therefore nondifferentiable at x = 0.

Possible flaws:

1) Not forcing b = 0 for the line leads to an ugly mess, and solving for m is algebra I would leave to wolframalpha.
2) I expect m to equal (2/3)(1/x)^(1/3) and I am missing the (2/3) in my writeup above. Why is this?

I do not understand that.

If that is an equation in x then you may say "assume these two are equal for some x. Let's find those x." Then the solution is obvious x = 0, y = 0, and is well defined.

If that is an equation in m then x needs to be specified.