An Epiphany

There is something strange going on in calculus. When we integrate

   xn 

we get xⁿ⁺¹/(n+1) which obviously does not work any more for n=-1.
We also know that the missing case n=-1 is covered by the natural log(x) function. How can it appear that a function from a completely different galaxy of functions appears so suddenly out of the blue (like a ``deus ex machina") or super nova?

In class on March 11, 2020, I told that an ``epiphany" explains the ``divine" emergence of the log from the more ``mundane" polynomials.

To see this, take the limit n → -1, of

   (xn+1-1
 -----------
    n+1

[ In class, I did not write down the -1 on the top. Thanks to Dylan for pointing this out! ]

This is a bit strange as we usually do not think about n as a variable. How do we take the limit? By bringing the function to the Hospital, of course! Let us take the limit

e(n+1) log(x)-1
------------------
    n+1

by differentiating nominator and denominator with respect to n (remember that n is the variable, not x). We get

e(n+1) log(x)  log(x) 
 -----------------
    1

Now, the limit n → -1, it is no problem to evaluate both top and bottom for n=-1 and we see that the result for n=-1 gives log(x). A new star is born!