Prove the limit of a near integer function by the epsilon-delta definition

Question

I have to prove the limits of the following function define as $f:\mathbb{R} \setminus\{2\} \to \mathbb{R}$. $$ f(x) = \frac{\left[ x - 2 \right]}{x - 2} $$

Limits to demonstrate: $$ \lim_{x \to 2^+} f(x), \quad \lim_{x \to 2^-} f(x) \quad \text{ and } \quad \lim_{x \to 2} f(x). $$

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I tried to demonstrate the first exists if $\forall\epsilon>0 \quad \exists\delta>0$ such as \begin{align} x < x - 2 < \delta &\implies \left| \frac{\left[x-2\right]}{x-2} - 0 \right| < \epsilon \\ &\implies \frac{\left[ x - 2 \right]}{x-2} < \epsilon \\ & \end{align} But I dont know how to manipulate the RHS to get the samething as the LHS.

Thanks for your help.

Answer 1

Hint: What is $[x-2]$ for x close to 2? (For example, 2.4?)