About the logic symbol $\exists$ I know that the symbol $\exists$ means there exists at least one
and that the symbol $\exists !$ means there exists unique

So if I want to say that the \$lim_{x \to a} f(x)$ exists, 

I can't write $\exists \lim_{x \to a} f(x)$ because this means there exists at least one limit
and I also can't write $\exists ! \lim_{x \to a} f(x)$ because it just doesn't seem right.
I know that both of these statements are correct. Though, is there another symbol for expressing this? Say, $\exists^t$ or something similar?
 A: Both statements are right, and since it's easy to prove that at most one limit exists, they are equivalent. So it's ultimately a cosmetic question. In cases like this, almost every text I've seen just uses "$\exists$" to not add an extra symbol. 
Something you might like better, though, is the following. Caution: this is basically never used in this context, I'm just mentioning it because you might find it interesting. Expressions like "$\lim_{x\rightarrow a}f(x)$" are really partial terms: they're like "$a+b$," except they're not always guaranteed to exist. In particular, it's built into their definition that they be unique. For partial terms, it's often more intuitive to think in terms of a new word, "is defined". E.g. "$a\over b$" is a partial term, and it's a little more helpful to say "$1\over 2$ is defined" than "${1\over 2}$ exists." I believe there are some texts which say "$\lim_{x\rightarrow a}f(x)$ is defined" instead of "exists," but that's rare. The symbol for this which is used in computability theory, and I believe elsewhere, is "$\downarrow$" for "defined" and "$\uparrow$" for "undefined"; so e.g. one would write $$\mbox{${1\over 2}\downarrow$ but ${1\over 0}\uparrow$}.$$ This way of thinking, I would argue, often matches better what we mean when we say things like "$\lim_{x\rightarrow a}f(x)$ exists" better. 
However, it is of course just a cosmetic change, and either "$\exists$" or "$\exists !$" does the job perfectly well; better, I would say, since it doesn't involve introducing a new symbol, and the payoff of "$\downarrow$" and "$\uparrow$" doesn't come until much later.
