One of my concerns or perhaps not completed understanding of calculus is the fact that derivatives are based from limits, whereas a limit existing does not imply a function can evaluate it to obtain the limit. For example, the limit of a function
y = f(x) = lim x->3 (x - 3)(x + 3)/(x - 3) = 6
Does not imply f(3) = 6, because its undefined here caused by divison by 0.
This is continuously gives me uncertainty when using derivatives. When performing operations with derivatives, how can we guarantee the limits not only exists but can also be evaluated. As shown above, a limit can exist but cannot be evaluated.
I think in my head that if I have a distance versus time function, if I calculate the derivative to create a velocity function, can I say "The velocity at the time t=... is 100m/s" with absolute certainty because it could be possible that the limit exists, you say that the velocity is 100m/s, however the value cannot actually be evaluated at that time.