Limit of the series of functions $f_n$

Question

$$f_n(x)= \begin{cases} 1&\text{if }\, x\geq 1/n\\ n|x|&\text{if }\, x< 1/n. \end{cases}$$ We want to find the pointwise limit of this function. I think the answer should be $f(x)=1\thinspace \forall x\in\mathbb{R}$. But the answer given is $$f_n(x)= \begin{cases} 1&\text{if }\, x\neq 0\\ 0&\text{if }\, x=0. \end{cases}$$

This is confusing me.

Answer 1

But clearly $f_n(0)=0$ for each $n\in\mathbb N$. Therefore, $f(0)=\lim_{n\to\infty}f_n(0)=0$.