Can a limit of function exist at a given point even if one of one-sided limits does not? Here are a few definition from my workbook:
Let $a\in\Bbb{R}\cup\{+\infty, -\infty\}$, $D\subset\Bbb{R}$. Then point $a$ is a limit point of $D$ if and only if there exists a sequence $\{x_n\}_{n \geq n_0}$ with terms in set $D - \{a\}$ such that $x_n \rightarrow a$.
And now we have a limit of a function $f: D \rightarrow \Bbb{R}$ where $a$ is a limit point of $D$ and $g \in \Bbb{R} \cup \{+\infty,-\infty\}$ defined as follows:
$g$ is a limit of $f$ at point $a$ if and only if for every sequence $\{x_n\}_{n \geq n_0}$ with terms in $D - \{a\}$ such that $x_n \rightarrow a$ we have $f(x_n)\rightarrow g$.
Now let's analyze function $f(x)=2$ defined at set $D=[0;1]$. We're analyzing point $a=1$. Clearly, we haven't got any right-sided limit there, as there doesn't exist any $d\in D$ which is greater than $1$. But according to these definitions, the limit at $1$ still exists and is equal to $2$ as for every sequence $x_n$ such that $x_n \rightarrow 1$ we have $f(x_n)\rightarrow 2$. Am I interpretting these definitions correctly?
 A: Depending on what situation you're in, it can either be convenient to say that there is a limit in such cases, or to insist that only the one-sided limit exists. Neither choice is inherently wrong, but of course it pays to be consistent in our use of words. (But beware that textbooks are not always consistent in this respect, especially between each other, so you may find some that assume one meaning and some that assume the other -- and they may not be explicit about their assumptions if the text uses the limit concept as a prerequisite rather than teach it from scratch).
The most common choice is to say that a limit does exist in this case, such that for example $\lim_{x\to 0}\sqrt x = 0$ without needing to specify a one-sided limit from the right. Formally we would take our definition of limit to be "for all $\varepsilon>0$ there is a $\delta>0$ such that for every $x$ in the domain of $f$ with $0<|x-x_0|<\delta$ it holds that such-and-such".
This choice has the pragmatic advantage that it is now easy to express the other concept when that is what we need -- we can simply say something like "assume that $f$ is defined in a punctured neighborhood of $x_0$ and $\lim_{x\to x_0}f(x)=L$" to get the more restrictive concept.
In contrast if the default meaning of "limit" included a requirement of $f$ always being defined close to $x_0$, it would be quite cumbersome to express the more lenient sense of limit when that is what is needed.
A: I think you are reading the definitions correctly. Since $f$ is defined only on D, there is no such thing as a "right-hand limit" at $1$. 
This means that the limit existing at $1$ is not a contradiction (nothing in the limit definition says anything about the right and left hand limits).
Similarly if a function is defined from $\mathbb{R}$ to $\mathbb{R}$, and you want the limit at $0$, then the limit of the continuation of that function along the imaginary axis in $\mathbb{C}$ is irrelevant.
