formal definition of limit to prove floor function limit

Question

$$\lim_{ x\to1−} ([|x|] − 2) = −2 $$

I'm trying to understand the proof of the following limit using a formal definition.

I know that;

$\lim_{x\to c−}f(x)=L$,means that given any $ϵ>0$, there exists $δ>0$ such that for all $x<c$, if $|x−c|<δ$, then $|f(x)−L|<ϵ$. but I'm not sure how I can handle the rest

Answer 1

Hint: for a floor function, take note of the following ($n \in \mathbb{N}$):

$\lim_{x \rightarrow n^{-}}\lfloor x \rfloor = n-1$

and on the other hand,

$\lim_{x \rightarrow n^{+}}\lfloor x \rfloor = n$.

Now, think of this from the perspective of the $\delta$-$\epsilon$-definition for a specific $\delta$. Then consider $|x-1|<\delta$. What follows from this?