# Is it axiom (implicit or explicit) that result of division (f(x) - f(x1)) / (x - x1) for x not equal to x1, is also true for x = x1? [closed]

Please let me ask a theoretical question. According to derivative definition, in accelerating motion (first) derivative at moment t1 is the limit of the function (constructed from position's function) for variable's limit t1. Then is it axiom that this function's limit is the speed at t1? Regards.

## closed as unclear what you're asking by Cameron Williams, Hans Lundmark, Mauro ALLEGRANZA, Shailesh, Brian BorchersApr 13 '18 at 14:09

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• Instantaneous acceleration is defined as the derivative of velocity. – Mauro ALLEGRANZA Apr 13 '18 at 8:13
• @ Mauro ALLEGRANZA, I mean Constant translational acceleration in a straight line (en.wikipedia.org/wiki/…) case [2]. Regards. – George Theodosiou Apr 13 '18 at 8:40
• velocity is defined as ratio of space and time and acceleration is dedined as ratio of velocity and time. When we consider them all function of time, we apply the rule of the calculus, and thus we can apply limits. – Mauro ALLEGRANZA Apr 13 '18 at 9:02
• @ Mauro ALLEGRANZA, Indeed velocity is found by taking the derivative of the position function: v(t) = x'(t). Regards. – George Theodosiou Apr 13 '18 at 9:40
• @ Cameron Williams, Hans Lundmark, Mauro ALLEGRANZA, Shailesh, Brian Borchers, I'm not in need, just asked that question for get know whether it is axiom. We accept that this limit (first derivative) is the speed at t1, but it is only limit by condition t never gets value t1, otherwise denominator becomes zero. What then is the speed at t1 and how we know that? – George Theodosiou Apr 13 '18 at 14:29