Q.A function is defined by $f(x)=x+5$ for all $x\neq 1$, and $f(1)=20$. Find $\displaystyle\lim_{x\to 1} f(x)$, and state whether it equals $f(1)$.

A limit reads the trend near the point, ignoring the value at the point itself. …

The limit $\lim_{x\to1}f(x)$ depends only on values of $f$ for $x$ near $1$ with $x\neq1$. For every such $x$, the rule is $f(x)=x+5$, so

The separately-assigned value $f(1)=20$ plays no part in the limit — a limit reports where the function is heading, not where it has been forced to sit at the single point $x=1$. …

Never substitute the specially-defined value $f(1)=20$ into a limit — the limit ignores $x=a$ itse …