Find the pointwise limit function of:
$$f_n(x)=\begin{cases}
0 & |x|> 1/n \\
nx+1 & x \in [-1/n, 0) \\
1-nx & x \in [0, 1/n]
\end{cases} $$
I think that in the limit, if we fix a certain $x$, we get that:
$$\lim _{n \to \infty} f_n(x)=\begin{cases}
0 & |x|> 0 \\
\infty & x \in [0, 0) \\
-\infty & x \in [0, 0]
\end{cases} $$
Where the second line is an empty statement, we rewrite this to:
$$\lim _{n \to \infty} f_n(x)=\begin{cases}
0 & |x|> 0 \\
-\infty & x =0
\end{cases} $$
Did I do this correctly? I'm not sure what the formal argument should be to get rid of the half-open interval $[0,0)$
Best Answer
Your answer is not correct. If $x=0$ then $f_n(x)=1$ for all $n$ and the limit is $1$. If $x \neq 0$ then $f_n(x)=0$ for all $n$ sufficiently large so the limit is $0$.