Why we can't take "dt" to 0 in a computer: Sources of error in numerical differentiation

We have seen that the error of numerical differentiation typically scales with the time step dt.  So why can't we just reduce the time step arbitrarily small to control the error?  This video describes how numbers are stored in a computer and how small roundoff errors are amplified by very small dt's, giving a lower limit to how small we can make our time steps.  

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This video was produced at the University of Washington

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0:00 Numerical roundoff error: How numbers are stored in a computer
8:40 Two sources of error: Roundoff and Taylor series