Let $$p_n(x)=a_n x^2+b_n x$$ be a sequence of quadratic polynomials where $a_n, b_n\in \mathbb R$ for all $n>0$.
Let $\lambda_0, \lambda_1$ be distinct nonzero real numbers s.t $\lim_{n\rightarrow \infty}p_n(\lambda_0)$ and $\lim_{n\rightarrow \infty}p_n(\lambda_1)$ exist.
Then:
1) $\lim_{n\rightarrow \infty}p_n(x)$ exists $\forall x\in \mathbb R$.
2) $\lim_{n\rightarrow \infty}p'_n(x)$ exists $\forall x\in \mathbb R$.
3) $\lim_{n\rightarrow \infty}p_n\left(\frac{\lambda_0 + \lambda_1}{2}\right)$ does not exist.
4) $\lim_{n\rightarrow \infty}p'_n\left(\frac{\lambda_0 + \lambda_1}{2}\right)$ does not exist.
Actually i think to consider $a_n$ and $b_n$ some sequence like $1/n$ and $n$ then to proceed but i am stuck