Rudin's Definition of the limit of a function confusion

Question

I was reading Baby Rudin, when I came across the definition he provides for the limit of a function, in terms of sequences. Namely, he says that for function $f : E \rightarrow \mathbb{R}$,

$$\lim_{x \rightarrow q} f(x) = p$$

if and only if for every sequence $x_n$ in $E$, where $x_n \rightarrow q$, $x_n \neq q$, we have $f(x_n) \rightarrow p$.

I don't understand what would motivate the condition that $x_n \neq q$. Any insight as to why we need this condition would be appreciated.

Answer 1

I don't understand what would motivate the condition that $x_n \neq q$. Any insight as to why we need this condition would be appreciated.

Say $f$ has a removable discontinuity at $q$, with $\lim\limits_{x\to q}f(x)=p,$ but $f(q)=r\ne p$.

Then if we allowed sequences $x_n$ with $x_n=q$, we could have $f(x_n)\to r$, not $p$.

Answer 2

For the same reason that $\lim_{x\to q}f(x)=p$ means$$(\forall\varepsilon>0)(\exists\delta>0):\color{red}{0<}|x-q|<\delta\implies\bigl|f(x)-p\bigr|<\varepsilon.$$Note that what happens when $x=q$ is irrelevant (since then $|x-q|\color{red}=0$). So, when we translate this definition into something equivalent using sequences, no term of those sequences can be equal to $q$.