Let $X$ be a $d \times d$ real matrix, $d>1$. Is it true that $$ e^X=\lim_{n \to \infty}\left(\text{Id}+\frac{X}{n}\right)^n\,\,\,? $$
Edit:
It seems that this question is a duplicate. To make it more interesting then, I am asking the following:
Can the density argument via diagonalizable matrices be made rigorous?
Write $g(X)=e^X, f(X)=\lim_{n \to \infty}\left(\text{Id}+\frac{X}{n}\right)^n $.
Then $f$ and $g$ agree on diagonalizable matrices (via the classical real-valued case). To deduce they coincide on all matrices, though, requires proving both functions are continuous. How do you prove that $f$ is continuous?
(I guess one might try to prove uniform convergence?)