Limits of positive polynomials

Let $(P_n)_n$ be a sequence of real polynomials and assume that $f(x):=\underset{n\to+\infty}{\lim}P_n(x)$ exists for all $x\in\R$.

1. Assume that $(P_n)_n$ converges toward $f$ uniformly on $\R$. Show that $f$ is a polynomial.

2. We do not make any assumption of the uniform convergence of $(P_n)_n$. Is $f$ a polynomial ? Is it a least $\mathcal{C}^{\infty}$ ?

3. Assume that the coefficients of $P_n$ are all non-negative for all $n\in\N$. Show that $f$ is $\mathcal{C}^{\infty}$.