Is this proof for Euler’s identity valid? $\DeclareMathOperator\cis{cis}$
I define $e^x$ to be
$$\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n,$$
so
$$e^{i\theta} = \lim_{n\to\infty} \left( 1+\frac{i\theta}{n} \right)^n.$$
Now I express $1+\dfrac{i\theta}{n}$ in polar form. The magnitude will be $$\sqrt{1^2+ \left( \frac{\theta}{n} \right)^2} = \sqrt{1+ \frac{\theta^2}{n^2} },$$
the argument will be $$\tan^{-1} \left( \dfrac{\theta/n}{1} \right) = \tan^{-1} \left( \dfrac{\theta}{n} \right).$$
So
$$e^{i\theta} = \lim_{n\to\infty} \left( \sqrt{1+ \frac{\theta^2}{n^2} } \left( \cis \left( \tan^{-1} \left( \frac{\theta}{n} \right) \right) \right) \right)^n.$$
By De Moivre’s theorem we get
\begin{align}
e^{i\theta} &= \lim_{n\to\infty} \left( \left( \sqrt{1+ \frac{\theta^2}{n^2} } \right)^n \left( \cis \left( n\tan^{-1} \left( \frac{\theta}{n} \right) \right) \right) \right) \\
&= \left( \lim_{n\to\infty} \left( \sqrt{1+ \frac{\theta^2}{n^2} } \right)^n \right) \cis \left( \lim_{n\to\infty} \left(n\tan^{-1} \left( \frac{\theta}{n} \right) \right) \right).
\end{align}
Using L’Hopital’s rule you can find
$$\lim_{n\to\infty} \left( \sqrt{1+ \frac{\theta^2}{n^2} } \right)^n = 1$$
and
$$\lim_{n\to\infty} \left(n\tan^{-1} \left( \frac{\theta}{n} \right) \right) = \theta,$$
so
\begin{align}
e^{i\theta} &= 1 \cdot \cis(\theta) \\ &= \cos\theta+i\sin\theta,
\end{align}
which is Euler’s identity. However, I am now questioning whether this is valid or not, as I can not find similar proofs online. This proof is the easiest to understand for me, so considering I can’t find much like it, it is probably wrong.
Any help would be appreciated.
 A: If I understand your posting, you are attempting to prove Euler's assertion that
$$e^{i\theta} = \cos(\theta) + i\sin(\theta). \tag1 $$
In fact, it is impossible for your attempt to be valid, because the underlying idea is not an assertion.  Instead, the underlying idea, as represented in (1) above, is nothing more than syntactic sugar.
You are starting from the premise that for any Real number $x$, that
$$e^x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n. \tag2 $$
The assertion in (2) above is absolutely true (i.e. has been proven), in Real Analysis, for any Real number $x$.  However, without some arbitrary definition (i.e. syntactic sugar), any expression of the form
$$e^{(x + iy)} ~: x,y \in \Bbb{R}$$
is gibberish whenever $y \neq 0.$
Therefore, the assertion can't be proven, because the assertion, in an of itself, is gibberish.
Euler decided to arbitrarily define
$$e^{(x + iy)} = e^x \times [\cos(y) + i\sin(y)].$$
Euler noticed that if he did this, that the exponent law (for example) :
$$e^{x_1} \times e^{x_2} = e^{(x_1 + x_2)}$$
still held, when complex numbers $z_1, z_2$ were substituted for $x_1, x_2.$
He then noticed that the behavior of certain limits, when the Real variable $x$ was replaced with the complex variable $z$ were well behaved.
So, the attack on certain Complex Analysis Math problems was facilitated.
However, the idea that you are attempting to prove, can't be proven.  That is, without the arbitrary definition, dipped in syntactic sugar, the idea is gibberish.
