It looks like we have
[tex]f(x)=\begin{cases}5-x&\text{for }x<5\\8&\text{for }x=5\\x+3&\text{for }x>5\end{cases}[/tex]
and we want to find [tex]\lim\limits_{x\to5}f(x)[/tex].
Since [tex]x[/tex] is approaching 5, we don't care about the value of [tex]f(x)[/tex] when [tex]x=5[/tex].
We do care about how [tex]f(x)[/tex] behaves to either side of [tex]x=5[/tex]. If [tex]x\to5[/tex] from below, then [tex]f(x)=5-x[/tex], so that
[tex]\displaystyle\lim_{x\to5^-}f(x)=\lim_{x\to5}(5-x)=5-5=0[/tex]
On the other hand, if [tex]x\to5[/tex] from above, then [tex]f(x)=x+3[/tex], so that
[tex]\displaystyle\lim_{x\to5^+}f(x)=\lim_{x\to5}(x+3)=5+3=8[/tex]
The one-sided limits do not match, since 0 ≠ 8, so the limit does not exist.
