The exponential function is a self-multiplication operation where is multiplied by itself

Until the 19th century, most mathematicians agreed, for reasons we will discuss below, that

## Is it equal to 1?

A clever student may cite this proof to prove this claim:

For

The student has proved it very slyly, for while the proof appears to be consistent, only someone scrupulous will tell you that it actually isn’t: after the substitution, we are dividing zero by zero, an operation that is meaningless or undefined. While it cannot be proved algebraically, can it be proved analytically?

Consider this trend:

The trend limns the function

However, this is not true. The function approaches *being exactly it*. This does not “prove” that

This, in fact, represents a neat proof for why

## It is undefined

There is no consensus today about whether the value is

There seems to be no answer either algebraically or analytically. We’ve only run into contradictions, just like one does when trying to determine what is zero divided by zero. The essential contradiction being: zero raised to any positive number must be zero, but any positive number raised to zero must be one; we cannot have it both ways. Mathematicians describe these problems as either meaningless or undefined.

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