Can anything equal DNE? I've come across several references where a person has shown a limit equal to DNE.  Something like $\lim_{x\to 0}\frac{1}{x}=DNE$. Is it ever reasonable to say that something is equal to something that doesn't exist, or has the person just used an inappropriate shorthand?
 A: Personally I would never write "$\lim_{x \to a} f(x) = \text{DNE}$", but if someone does write that (on an exam solution or something) it is just a sloppy way of saying "the function $f$ does not approach a limit as $x$ approaches $a$."
A: restatement of, and direct answer to, the Question:
Let $c:= \mathtt{«}\text{an expression with }x\mathtt{»} \quad$and$\quad \mathtt{DNE}:=\mathtt{«}\text{an arbitrary nonexistent object}\mathtt{»}$.
Then, from that, the answer to

Can $c=\mathtt{DNE}$?

$\quad$ is, technically, Yes, iff you make adequate logical justifiable assumptions (but No if you faultily allow some loose logical construction, or provide no reasoning at all), for the cases when $c$ does not exist. Sufficient logical justification is possible but lengthy (and moreover most likely skipped and hence not recognized or simply considered notationally wrong, since nonstandard), whereas an alternative notation is much more automatically understandably correct.

Logical justification of the affirmative:
Let $\mathtt{«}\text{an arbitrary nonexistent object}\mathtt{»}:=d$. Then by transitivity, $\mathtt{DNE}:=d\:=:c$.
But how is $d$ defined?
By definition, $∄d$. In a manner of speaking, this
$\quad ⟹\quad d∈\{\}\:⇔\:∅∋d\quad \quad $which seems to contradict the literal construct defining the nullset as empty i.e. having no members. This could be justified by recognizing as, and proceeding to define to be, $d$ as ‘null_element’; parallel to how $∅$ is defined to be the ‘null_set’.$\:\:\:$ In light of this sort of leap in logic, I would propose a different symbol for '$d$' as used herein: the lowercase slashed-o ($ø$), since the uppercase version ($Ø$) looks typographically similar to the emptyset symbol.$ \:\: $Of course, this still leaves a gap in converting “DNE” to “ø”, where ‘DNE’ is taken to be an object of some sort rather than a qualitative state such as a group action.

further considerations, with possibly better formal notation:
Read aloud in non-abbreviated form, "DNE" as used here (a limit that doesn't converge ergo doesn't exist) is simply its de-initialism “Does Not Exist”, rather than “Non-Existent Entity” or the like. As such, "$c=\mathtt{DNE}$" ought to be read as “c equals does-not-exist” or “c is equal to does_not_exist”, which is ungrammatical. On the otherhand, one could simply slap the nonexistential quantifier in place of the equals sign & RHS, and call it a day: "$c\:∄$". Problem with this is that placing quantifiers after instead of before the modified phrase is non-standard, especially in a standalone expression not contingent on something else. Thus instead "$∄c$" would be more notationally correct, despite its following a different “grammatical form” than in the existent-type case (since if $∃c$, then unless you don't yet know the existent value then you would more likely want to indicate a less-vacuous $c=\mathtt{«}\text{its value[s]}\mathtt{»}$). Considering this, it would make more linguistically and notationally valid sense to replace ‘$=\mathtt{DNE}$’ with ‘$\:\mathtt{DNE}$’ (omitting, or for typographic clarity instead substituting with a space or two, the ‘$=$’).
If you would like to retain the general flow of the mathematical statement in the does-not-exist as in the does-exist[-and-equals-«blah»] case, but think that following the expression by the freestanding letters “DNE” is aesthetically displeasing, then you could probably be able to convey the meaning (in consistent form) by swapping the equality-symbol with some other relational symbol and an appropriate glyph or glyphs to the right of that. I’m not sure what the most technically correct & appropriate option would be (and would appreciate Comments on the matter), but perhaps an equivalence relation (‘≡’) or strong_entailed-by(‘⫤’) could do the trick, such as “$c≡\mathtt{DNE}$” or perhaps “$c⫤∅$”.
