# Math 1a Spring 2020

## 1a Introduction to Calculus

# An Epiphany

There is something strange going on in calculus. When we integratex^{n}

^{n+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

(x^{n+1}-1 ----------- n+1

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

e^{(n+1) log(x)}log(x) ----------------- 1