$f:\mathbb R \to \mathbb R$
$$f(x)= \lim_{n \to \infty} \left[ \frac{n^n(x+n)(x+n/2)...(x+n/n)}{n! (x^2+n^2)(x^2+n^2/4)...(x^2+n^2/n^2)} \right] ^{x/n}$$
Find the function f.
The way I evaluated the limit is by taking out a common factor of $n$ or $n^2$ from each bracket.
$$\lim_{n \to \infty} \left[ \frac{n^{2n}(\frac{x}{n}+1)(\frac{x}{n}+1/2)...(\frac{x}{n}+1/n)}{n!*n^{2n} (\frac{x^2}{n^2}+1)(\frac{x^2}{n^2} +1/4)...(\frac{x^2}{n^2} +1/n^2)} \right] ^{x/n}$$
But $x/n \to 0$ so this simplifies to $$\lim_{n \to \infty} \left[\frac{1/2*1/3....*1/n}{n! * 1/4 * 1/9 * ... *1/n^2}\right]^{x/n}$$ $$\lim_{n \to \infty} 1^{x/n} = 1^0 = 1$$
But this is wrong. I want to know where the mistake is, a rigorous/formal way to point out of my mistake.
Note: I do not want the solution.