$limlimits f(x)_{x to c}$ exists for all values of c except for at $c=4$

A limit is the $y$ value of a function as it gets infinitely close to an $x$ value. A limit exists if $lim limits f(x)_{xto c^{-}}$ $=$ $lim limits f(x)_{xto c^{+}}$. When finding limits on a piecewise function, you first check to make sure each individual 'piece' exists at the given bounds. Then you make sure that at the endpoints of each boundary, the two functions equal each other. In this problem, $limlimits f(x)_{a to 2^{-}}=4$ and $limlimits f(x)_{a to 2^{+}}=4$. However, at the other boundary $x=4$, $lim limits f(x)_{xto 4^{-}}= 0$ and $lim limits f(x)_{xto 4^{+}}=4$, so the limit does not exist at $x=4$