TITLE: Existence of a limit or misspelled.
QUESTION [6 upvotes]: If someone proposes the problem:
Calculate the limit: $\lim_{x\rightarrow 2}\frac{x-2}{2-\sqrt{4}}$
For me the limit does not exist because in fact the function $\frac{x-2}{2-\sqrt{4}}$ does not exist. However, it is also true that the problem is misspelled. But my question is that if they already pose the problem to you like this, what is the correct thing to say, you cannot write such a limit (misspelled) or it does not exist.

REPLY [3 votes]: I would say that the expression is not well defined because it is equivalent to $\frac{x-2}0$ and therefore it is meaningless consider the limit at $x=2$ which is not a cluster point.
We could define a value $L$ for the function at $x=2$ and then extend the notion of limit to isolate points such that
$$\lim_{x\rightarrow 2}\frac{x-2}{2-\sqrt{4}}=L$$
as discussed in the following references

Limit of a function as approaching an isolated point of its domain?
Why not define 'limits' to include isolated points?