Edit] Using the limit definition

The exponential function e^z can be defined as the limit of (1 + z / n) ⁿ, as n approaches infinity. In this animation, z = iπ /3, and n takes various increasing values from 1 to 100. The computation of (1 + z / n) ⁿ is displayed as the combined effect of n repeated multiplications in the complex plane. As n gets larger, the points approach the complex unit circle (dashed line), covering an angle of π /3 radians.

An alternative proof ^[6] starts from the limit definition of e^z:

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Plug in z = ix, and let n be a very large integer. Then consider the sequence:

(The last element of the sequence approaches e^ix.) If the points of this sequence are plotted in the complex plane (see animation at right), they roughly trace out the unit circle, with each point being x / n radians counterclockwise of the previous point. (This statement is more and more and more accurate as n increases. The proof is based on the rules of trigonometry and complex-number algebra. ^[6] ) Therefore, in the limit the last point in the sequence, (1 + ix / n) ⁿ, is the point on the unit circle of the complex plane located x radians counterclockwise from +1, i.e. the point cos x + i sin x. Therefore, e^ix = cos x + i sin x.