Let us consider a sequence $(p_l)_l$ of polynomials on $[0,1]$ that converge uniformely, as $l\to \infty$, to a function $f$ defined on $[0,1]$.
I denote the polynomials $p_l(t) = \sum_{k=0}^{m(l)} a_k(l) t^k$.
I have also a certain quantity $\lambda(p_l) \in \mathbb{R}$ which has the following asymptotic behaviour :
$\lambda(p_l)(x) = \sum_{k=0}^{m(l)} a_k(l)\tfrac{1}{1+k/x} \underset{x\to\infty}{\sim} \sum_{k=0}^{m(l)} a_k(l) \sum_{j=0}^{\infty}(1)^j (\tfrac{k}{x})^j$.
So:
$\lambda(p_l)(x) \underset{x\to\infty}{\sim} \sum_{k=0}^{m_l}a_k(l)  \sum_{k=0}^{m(l)} a_k(l)\tfrac{k}{x} + ... = p_l(1)  \tfrac{p_l'(1)}{x} + ...$
Naively, I want to apply the limit to obtain:
$\lambda(p_l)(x) \underset{x\to\infty}{\sim} f(1)  \tfrac{f'(1)}{x} + ...$
Questions : Do I have first check some hypothesis to write it? Do you have any reference about applying the limit in an asymptotic behavour?
Thank you a lot.



Note that the quantity $\lambda p$ depending on the parameter $x$, as you define it for polynomials $p(t):=\sum_{k=0}^m a_k t^k$, that is
$$(\lambda p )(x)=\sum_{k=0}^m a_k \frac{1}{1+k/x}\, , $$
can be extended to a positive, bounded linear operator
$$\lambda :C^0([0,1])\to C^0_b([0,\infty))$$
taking the function $f\in C^0(I)$ to $$*$$ You can also write, changing variable in the integral, ($t=e^{\tau}$)
$$(\lambda f)(x):=x \int_0^{+\infty} e^{\tau x}f(e^{\tau})d\tau\, .$$ 

