Let $g_n:[0, \infty)\to \mathbb R$ be a differentiable functions, and $|g_n(n)| \geq 1$ for $n\in \mathbb N.$ Assume that $\int_0^{\infty} |g_n(t)|^6 t^2 dt < \infty$ for each fixed $n.$ ...(A)
Assume that $I_n=\int_0^{\infty} |g'_n(t)|^2 t^2 dt < \infty$ for all $n\in \mathbb N.$
Define $h_n(n):= (g_n(n+1)-g_n(n))^2, n\in \mathbb N$
Question: (1) Can we expect $I_n \to \infty$? (2) What can we say about the behavior $h_n$ at $\infty$? Can we say $\lim_{n\to \infty} h_n(n) >c$ for some $c>0$?
Note: (1) For instance, $f_n(x)=e^{(n-x)^2}$ satisfies the above situation. (2) If we drop condition $(A)$, then the answer to (1) is negative. (see this)