Limits on functions that have points of non-existence. I am currently teaching myself rigorous calculus from Spivak's Calculus. I noticed that the definition of a limit prevents the point at which the limit is being considered from being mentioned. However, no mention is made of choosing a delta such that there is an x within that delta where a function is defined. Is there a precise way of defining how to avoid this?
For example, what about a function on the real numbers that does not exist anywhere. My intuition tells me that no limit exists on this, but what about other functions with periodic points at which the limit does not exist?
 A: One typical definition of a limit is the following:

If $f:M \longrightarrow N$ is a function from a subset $M$ of a metric space $A$ to a subset $N$ of a metric space $B$, we usually say that $\lim_{x \to c} f(x) = L$ if for every neighborhood $V$ of $L$ in $N$, there is a neighborhood $U$ of $c$ in $M$ such that $f(U - \{ c \}) \subseteq V$. Or equivalently, for every $\epsilon > 0$, there is a $\delta = \delta(\epsilon) > 0$ such that for all $x$ within $\delta$ of $c$ in $M$, $f(x)$ is within $\epsilon$ of $L$ in $N$.

Let's consider the function $f: x \mapsto x$ defined on the subset $M$ of the real line $(-\infty,1]\cup\{2\}\cup[3,\infty)$. What is $\lim_{x\to 2} f(x)$?. Well, it's anything we want. No matter what we claim $L$ to be, we might choose $\delta = 1/2$, and the empty set is certainly within any neighborhood $V$ of $L$. 
This is annoying and dissatisfying. So more pedantic definitions of a limit also require that $c$ be an accumulation point of $M$, so that there are sequences of numbers actually going to $c$. This is the best way that I know of to avoid that sort of trivial error.
However, your last question is a bit different. In essence, you are asking about the function $f: \emptyset \longrightarrow \mathbb{R}$. This function is continuous, but no limits exist because a limit is a statement of behaviour near a point $c$, and there is no $c$ in the empty set.
