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"Irrational" limit behavior

Near the end of August, Maxim noted a limit which is clearly not 0, yet
which Mathematica claims to be 0 . It had the form

  Limit[x - Sqrt[Pi], x -> q]

where q was merely a very close rational approximation of Sqrt[Pi].

Here's a similar example:

In[1]:= Rationalize[Sqrt[E], 10^(-74)]

In[2]:= Limit[x - Sqrt[E], x -> %]
Out[2]= 0

I find this "irrational" behavior to be extremely disturbing. Please,
someone, explain to me what's going on!

David Cantrell

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