String.raw is used with template literals, the new(ish) ES6 syntax for strings with interpolated variables, embedded quotes and newlines, and the optional application of "tags" (which are just function callbacks that receive the pieces of the template split up by each interpolated variable to return the final combined string however you like).
So if I have a function that processes a template literal, and it's called processLiteral, then I could do something like this to apply it to a literal:
processLiteral`This is a string with interpolated variables like ${someVar} and also ${otherVar}.`
Specifically, String.raw is meant to be used as the tag of a template literal, and it will return the raw string without processing any escaped characters. Meaning \n will remain as \n and not become a newline, \t will remain as \t and not become a tab, etc. It does this by automatically adding slashes before any escaping slashes, thereby escaping them, so it's equivalent to using \\n, \\t, etc.
So in your case, if you want to use that method, since you can't put template literals directly in an object key declaration (it doesn't parse correctly), you'd do something like this:
var Array_QA = {};
Array_QA[String.raw`\Large {\lim_{x \to 2}} (5^x + 2^x + 4)`] = "33";
Array_QA[String.raw`\Large {\lim_{x \to 2}} \frac {x^2 - 6x + 8}{x^3 - 4}`] = "0";
That said, having complex strings like that as object keys is probably not a good idea from the start... but it should work. I'd suggest refactoring your code to do that differently, if you can.