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 ::You can cancel out zeroes, which is why the limits are what they are. The differential equation I'm referring to is the basic slope formula or whatever other areas call it. When you use f(x) instead of y, the equation is dy/dx = (f(x+h) - f(x)) / h. This is the equation used to find the slope of a line between two points. When this is done with any equation with a leading x power > 1, you result has xs and hs. To determine the ""instantaneous veloctiy"", ""slope"", whatever you want to call it, 0 is substituted for every h value because you want the slope at that point. Now this part is crucial. Slopes work, and they are proven to work, otherwise we wouldn't use them because they wouldn't work. To say that you can't divide by 0 makes the whole beginning equation moot. And yes, the definition of refute is ""to prove to be false or erroneous, as an opinion or charge."" You would be refuting the whole ability to find the slope on a given point, which is a significant part of calculas. I know you will rant about ""Slopes work, and they are proven to work, otherwise we wouldn't use them because they wouldn't work."" but there is no other way to word it. It's like telling me to prove adding works. It works because it does, it doesn't need a reason."