When we take the limit towards somthing, do we never actully attain the number?
For example,
Limit involving greatest integer function and modulus
They seams to exploit that we never hit zero and that they thus have,
$\lim_{x \rightarrow0} \lfloor \cos x \rfloor=0$
Sometimes we do, sometimes we don't. We don't require non-attainment, we only require that we can get as close as we want to that value as long as we go "close enough" to the limit value.
If $\lim_{x \rightarrow a} f(x) = b$, it means that if I ask, "can you make $f(x)$ get "this" close to $b$?" You say "yes, I can" and then provide an $x$ that is "close enough" to $a$ such that $f(x)$ is that close to $b$, no matter how close I asked you to get $f(x)$ to $b$.
That is, for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $|x-a|< \delta$, then $|f(x)-b| < \epsilon$.
Note that here in relation to the above informal explanation "$\epsilon$" represents how close I asked you to get to the function, and "$\delta$" represents your response, "I just need to be this close to $a$."