Intuition for Euler's formula by using the limit $\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n$? My goal is to give good intuition as to why the formula
$$e^{i\theta}=\cos\theta+i\sin\theta$$
Is correct. I don't need to be formal or rigorous.
One good way of doing this is by looking at the taylor expansion of $e^x$ for real values of $x$, substituting $x=i\theta$ and getting the taylor expansions for $\cos$ and $\sin$.
What I want: To give a similar intuitive explanation without using taylor expansions.
What I do have: The equality
$$e^x=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n$$
So, I can ask myself what happens with this formula when I substitute $x=i\theta$. But I'm not sure what happens now - can we really proceed from this starting point to get an intuition for Euler's formula?
 A: You could denote $\mathrm{cis}\theta=\cos\theta+i\sin\theta$, and show (with induction and trigonometric angle addition formulas) that $\mathrm{cis}(n\theta) = (\mathrm{cis}\theta)^n$. (Showing first that $\mathrm{cis}(\theta+\phi)$ = $\mathrm{cis}\theta\cdot\mathrm{cis}\phi$ may or may not be easier.)
Then use the fact that for small $\theta$, $\sin\theta\approx\theta$ and $\cos\theta\approx 1$ to find
$e^{i\theta}\approx\left(1+\frac{i\theta}{n}\right)^n\approx\left(\mathrm{cis}\left(\frac{\theta}{n}\right)\right)^n=\mathrm{cis}\theta=\cos\theta + i\sin\theta$, where the approximation becomes exact as $n\to\infty$.
A: In order to have a better understanding of Euler’s formula, you should picture yourself the two-dimensional complex plane. In fact, the right side of the equation describes the circular motion along the outside of the unit unit circle using sine and cosine. That is, if we set $\theta = \pi$, then that means that we travel $\pi$-units along the unit circle. We are then half way around the circle, i.e. we’re at $-1 = e^{i\pi} = \cos(\pi) + i\sin(\pi)$.
