1.9, Euler's Marvelous Formula!

Long ago (about 1740) Euler discovered that if x is fixed then as n goes to infinity

(1+x/n)^n goes to e^x.

cos(x/n) goes to 1

and

sin(x/n)/(x/n) goes to 1.

He then rewrote DeMoivre's formula

(cos z + i sin z)^n = cos nz + i sin nz

as 

(cos x/n + i sin x/n)^n = cos x + i sin x

He noted that sending n to infinity does not effect the right side of this equation but does effect the left side and so, "letting n be infinite" (!) he said that the lefthand side of this equation was

(1+i x/n)^n

But then, as n goes to infinity, that is just e^ix. So
e^(ix) = cos x + i sin x!

This remarkable result can be proven in other ways  – I like the Taylor series proof, which Euler came up with later – but most important are the many applications of this identity.  This identity first explains that trigonometric functions and exponential functions are just different views of the same mathematical object.

Additional resources

My notes on Euler's Marvelous Formula are in my Google Drive folder.  A precalculus lecture on Euler's formula is available here.

On my Google Drive folder for precalculus are notes on complex numbers and the polar coordinate form of complex numbers.

Some other notes on the web about complex numbers are here and here.