In order to determine a result, it's true that dx can't be 0, but paradoxically, the computations with dx!=0 aren't (necessarily) the expected solution either and which way this paradox is resolved relies upon subjective physical and geometric assumptions. Notice that in order to determine the slope of a line segment, you need two distinct points. The idea of passing a tangent line through a curve at only a single point is a logically impossible structure as there's no way to compute the slope of a single point.

As a quick example, if we have a function defined at only 3 points, which linearly ordered are A, B and C then only 2 slopes exist between these points, the segments AB and the segment BC, and in general, a function with n solutions only has n-1 derivatives. Also notice that the derivative at B is not precisely defined as this point is an endpoint for two different line segments. (A more deterministic manner of defining a slope would be, for example, to associate a derivative with the line segment to the left of a point, for example, but this is similar to retaining information regarding the quantity dx and doesn't provide the simplification desired for infinitesimal calculus and you still have a point that's an offset or baseline for the function, with no derivative for it - notice that the inverse operation, integration, requires that an unknown constant be added 'C').

Infinitesimal calculus relies upon appending physical and geometric assumptions of smallness, nearness and other forms of potential insignificance to quantize something with an assumed minimal impact on a solution into nothingness, but this quantization is paradoxically something not insignificant as the final step is to remove the record of its existance - something that can't be approached gradually.

Also notice that, though it's obvious that dx must "approach" zero in order for someone to assume the properties could be applied for dx=0 also (notice the elements of indeterminism here as the concept relies upon potentially incorrect and illogical assumptions based upon physical and geometric analogies), but that the manner in which one such quantity, approaching zero, did so relative to another such quantity approaching zero, is not generally considered and this leaves many indeterminant and subjective interpretations to results derive from infinitesimal calculus (not also that finite calculus skips these indeterminant steps and works within finite spaces to determine precise results).