I had a question about the definition of a limit. I know for a limit to exist the right hand limit must equal the left hand limit but what if the graph of a specific function has the domain from [0,5] and if you wanted to find the limit at x=5 you know the left hand limit exists but the right hand doesn't.
So in this situation does the limit exist or not since only the right-hand limit exists?
On
Check out the usual definition of the limit of a function of a real variable (https://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit#Precise_statement).
If $D$ is any subset of $\mathbb{R}$, and $f: D \rightarrow \mathbb{R}$ is a function, and $c \in D$ (more generally $c$ can be a limit point of $D$), then we can talk about whether $\lim\limits_{x \to c} f(x)$ exists.
This stuff about left and right hand limits needing to coincide for the limit to exist only applies when $c \in D$ and $D$ contains an open interval about $c$.
For example, suppose $D = [0,5]$, and $f: D \rightarrow \mathbb{R}$ is the function $f(x) = x^2$. Then $\lim\limits_{x \to 5} f(x)$ exists and is equal to $25$, because for any $\epsilon > 0$, there exists a $\delta > 0$ such that if $x \in D$ and $|x - 5| < \delta$, then $|x^2 - 25| < \epsilon$.
So in this case, $\lim\limits_{x \to 5^-} f(x)$ and $\lim\limits_{x \to 5} f(x)$ exist, but it makes no sense to talk about $\lim\limits_{x \to 5^+} f(x)$.
You can only discuss limits in neighborhoods where the function is defined.
If the function is only defined in $[0, 5]$, then it makes no sense to talk about a right-hand limit at $5$. Therefore, when you talk about a limit at $5$, you can only mean left-hand limit.