you just keep on babbling. but you don't have a proof. you have a result but you don't know how you arrived at the result."Infinity does not exist" for division by zero, because zero is a point, not a length, and the real numbers (even the integers) requires a "distance"; i.e., a metric. Without a metric, the number line doesn't exist (is irrelevant; if there are elements, it is an unordered set. Zero only exists as the midpoint of all possible lengths (or as the endpoint of all possible negative or positive lengths)
John is wrong because he does not understand the concept of independent variables; e.g., f(x,y) = x/y ; that is, two dimensions in Cartesian coordinates. Since he doesn't understand that, he doesn't understand curvature, in terms of which is defined as the ratio of the circumference to the diameter =(2r)/(2r). Since this is a fraction, it requires two dimensions for definition.
And his definition of slope is FLAT wrong, unless m = 0, even as an misguided attempt at differential geometry or finite element analysis... And even if m = 0, the slope is correct, but a line (a chord) is all he's got - the derivative is the rate of change along a curve at an arbitrary point x, where the curve is a function of x, and the chord is arbitrarily small. That is, by the mean value theorem, one knows that there will be a parallel line a line specified by the chord, but it will not be known at which point x the chord touches the function at a single point, since for the derivative, the function can be arbitrary.
"Existence" of infinity as a mental concept is all in the mind, like the existence of god - one can either believe or not, but an intellectual justification is a fools' errand; a discussion among idiots.... (well, ok, the mathematically challenged...
for example you have
without knowing how it is possible