The tidiness of elementary mathematics
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Version vom 9. November 2011, 00:18 Uhr von Sangwinc (Diskussion | Beiträge)
We all expect a certain tidiness to mathematics, and for many this provides a gratifying aesthetic pleasure. However, there are some wrinkles. One of these arises in the integral of . Calculus provides us with the formula
However, this equation is only correct for
. If we try to take
the right hand side is meaningless because we have a zero on the denominator of the fraction, i.e.
. But in this case a separate argument gives the answer
Our expectation is that these two formulae should be reconciled. Indeed, if we let
approach
in the first we should end up with the second, but that fails to happen. For each
,
is undefined. You can try this on the graph below.