Is there a symbol for ‘equal if defined’ Can anybody recommend a symbol for ‘equal if defined’ as an asymmetric concept?
In contexts where one might write down notation for an undefined quantity (such as $1/x$ when $x$ might be $0$), sometimes people use the equality symbol to mean that if either side is defined, then so is the other, and then they are equal; and sometimes people use it to mean that if both sides are defined, then they are equal.  Sometimes they use this together with a variant symbol with the other meaning.  These are not the meanings that I want.
I want a symbol that says that if the left-hand side is defined, then so is the right-hand side, and then they are equal.  Or a symbol that means that if the right-hand side is defined, then so is the left-hand side, and then they are equal.  Either way, the meaning is asymmetric.  Preferably a symbol that is itself left-right asymmetric, so that the reverse symbol has the reverse meaning.
Usage examples:  In Algebra, when we write that $(x^2 - 1)/(x^2 - x) = (x + 1)/x$, what we really mean is that if $(x^2 - 1)/(x^2 - x)$ is defined, then $(x + 1)/x$ is also defined and then $(x^2 - 1)/(x^2 - x)$ is equal to $(x + 1)/x$.  However, if $(x^2 - 1)/(x^2 - x)$ is undefined, then we're not claiming that it's equal to anything, and we're also not saying whether $(x + 1)/x$ is defined.
In Calculus, when we write that $\lim_{x \to c}(f(x) + g(x)) = \lim(f(x)) + \lim(g(x))$, what we really mean is that if $\lim(f(x)) + \lim(g(x))$ is defined, then $\lim(f(x) + g(x))$ is also defined and then $\lim(f(x) + g(x))$ equals $\lim(f(x)) + \lim(g(x))$.  However, if $\lim(f(x)) + \lim(g(x))$ is undefined, then we're not claiming that anything's equal to it, and we're also not saying whether $\lim(f(x) + g(x))$ is defined.
Of course, one can always write ‘if the left-hand side is defined’ after the equation, or something like that.  But I want a symbol that I can use with a string of conditional equalities (all in the same direction) in the course of an argument to establish an overall conditional equality.  Example: $\lim_{x \to 5}(x^2 + 6) = \lim(x^2) + \lim(6) = \lim(x)^2 + 6 = 5^2 + 6 = 31$, where at each stage I use a basic rule of limits in the course of the calculation.  Ultimately, I conclude that if $31$ is defined (which it is), then $\lim(x^2 + 6)$ is defined and equals $31$ (which it does).
Of course, I can make up a symbol, but if anybody has already made one up and used it successfully (either in a formal logical context or informally as I was doing above), then I'd like to hear about that.
 A: As far as functions go, note that we already have a symbol for this (coming from set theory): "$\subseteq$"! If $f$ and $g$ are partial functions, identifying them with their graphs means that we can write "$f\subseteq g$" to mean "$f(x)=g(x)$ whenever $f(x)$ is defined."

More generally (or in contexts where imposing set theory might be undesirable), here's a possible recommendation:
First, the symmetric version. I don't know how common this is elsewhere, but in computability theory, we often write $$x\simeq y$$ if $x$ and $y$ are expressions which are either both undefined, or both defined and equal. (Occasionally "$\cong$" is used instead of "$\simeq$," but I prefer "$\simeq$" since (at least within computability theory) it's less often used for isomorphism.)
For the asymmetric version, I would recommend "$\gtrsim$" in analogy with "$\simeq$." But I haven't seen that before, so I don't actually know if it's commonly used; it's just what seems to me the natural choice.
A: Well, I think that the answer seems to be No, that there is no such symbol in standard use in any context.  More's the pity!  Perhaps we can invent one.
A: Motivated by Toby's answer and his accompanying comments (especially the second one), perhaps a little modification like the following can make clearer which side the "if defined" applies to. We can write $$L \mathop{{}^\downarrow{=}} R$$ to mean "if $L$ is defined then $L=R$". For example, $$\frac{x^2−1}{x−1} \mathop{{}^\downarrow{=}} x+1$$
Similarly, we can write $$L \mathop{{=}^\downarrow} R$$ to mean "if $R$ is defined then $L=R$". For example, $$\lim (f(x)+g(x)) \mathop{{=}^\downarrow} \lim f(x) + \lim g(x)$$
The choice of using $\downarrow$ is from this comment from another question about the symbol for "defined". Another choice would be $\vartriangle$ (that is "$L \mathop{{}^\vartriangle{=}} R$" and "$L \mathop{{=}^\vartriangle} R$") which perhaps work better for those already familiar with $\triangleq$.
Personally I like $\downarrow$ more though, since it is still visible but takes up less space.
