[tex]f(x)=\begin{cases}5-x^2&\text{for }x<0\\-5&\text{for }x=0\\2x+5&\text{for }x>0\end{cases}[/tex]
[tex]\displaystyle\lim_{x\to0^-}f(x)=\lim_{x\to0}(5-x^2)=5[/tex]
[tex]\displaystyle\lim_{x\to0^+}f(x)=\lim_{x\to0}(2x+5)=5[/tex]
[tex]\implies\displaystyle\lim_{x\to0}f(x)=5[/tex]
- - -
[tex]f(x)=\begin{cases}x-3&\text{for }x<4\\x+3&\text{for }x>4\end{cases}[/tex]
[tex]\displaystyle\lim_{x\to4^-}f(x)=\lim_{x\to4}(x-3)=1[/tex]
[tex]\displaystyle\lim_{x\to4^+}f(x)=\lim_{x\to4}(x+3)=7[/tex]
[tex]\implies\displaystyle\lim_{x\to4}f(x)\text{ does not exist}[/tex]
