Limit of sequence and limit of function sequence

Question

Suppose that for a function $h(x)$, there is a sequence $x_n$ such that $\lim_{n \to \infty}h(x_n)=h(c)=\min_x h(x)$. My understanding is $\lim_{ n\to \infty} x_n \neq c$ in general. But can somebody provide examples to verify the inequality?

Answer 1

Simplest counterexample I could come up with: take $h\colon\mathbb{R}\to\mathbb{R}$ be the identically zero function, i.e., $h(x)=0$ for all $x$.

Take any $\ell\in\mathbb{R}$, and any sequence $(x_n)_n$ converging to $\ell$. Then $$ \lim_{n\to\infty} h(x_n) = h(\ell) = 0 = h(\ell+1) = \min_x h(x) $$ but clearly $\ell\neq \ell+1$.

Answer 2

Let $h(x)=x^2e^{-x^2}$, then $\min_{x\in\mathbb{R}} h(x)=0$. If we take $x_n=n$ we get $\lim h(x_n)=0=h(0)$, but $x_n\not\to 0$.