I have to provide a proof for a function but I'm struggling to grasp the main concept.
$$\lim_{x \to 3} \left\lfloor \frac{x}{2}\right\rfloor = 1 $$
Here is what I've come up with: $$\frac{x}{2} - 1 \lt \left\lfloor \frac{x}{2}\right\rfloor \lt \frac{x}{2} + 1$$
But I'm stuck from here since I cannot use anything else but $\varepsilon$-$\delta$ (meaning no squeeze theorem.) Each of the limits (for every side) is not helpful. I get: $$ \frac{1}{2} \lt \lim_{x \to 3} \left\lfloor \frac{x}{2}\right\rfloor \lt \frac{5}{2}$$
Any clarifications are welcome! Thanks!