We had discussed this on Nov 17 2015 blog. We finish the article with more examples from the work of Prof. Terence Tao. (If you like it, please send a thanks to him :-))
Example 1. (Interchanging limits and integrals).
For any real number y, we have
Taking limits as , we should obtain
But, for every x, have . So, we seem to have concluded that . What was the problem with the above argument? Should one abandon the (very useful) technique of interchanging limits and integrals?
Example 2. Interchanging limits and derivatives.
Observe that if , then
and in particular that
Taking limits as , one might then expect that
But, the right hand side is . Does this mean that it is always illegitimate to interchange limits and derivatives?
Example 3. Interchanging derivatives.
be the function . A common manoeuvre in analysis is to interchange two partial derivatives, thus one expects
But, from the quotient rule, we have
and in particular,
On the other hand, from the quotient rule again, we have
Since , we thus seem to have shown that interchange of two derivatives is untrustworthy. But, are there any other circumstances in which the interchange of derivatives is legitimate?
We are familiar with the beautifully simple rule
but one can still get led to incorrect conclusions if one applies it incorrectly. For instance, applying it to , and we would obtain
But this is an incorrect answer since .
Of course, all that is going on here is that is only applicable when both go to zero as , a condition which was violated in the previous example. But, even when and do go to zero as , there is still a possibility for an incorrect conclusion. For instance, consider the limit
Both numerator and denominator go to zero as , so it seems pretty safe to apply the rule, to obtain
The first limit converges to zero by the Sandwich theorem (since the function is bounded above by and below by , both of which go to zero at 0). But the second limit is divergent (because goes to infinity as , and does not go to zero.) So the limit diverges. One might then conclude using that also diverges; however, we can clearly rewrite this limit as , which goes to zero when by the Sandwich Theorem again. This does not show that is untrustworthy. Indeed, it is quite rigorous, but it still requires some care when applied.
That is all, once again, if you like this, please send a thanks note to Prof. Terence Tao.