The reason why we could choose a bound of $\mid x - 2\mid$ is that the limit is a local feature. We could only consider the case where $x$ is near $2$.
How we choose $\delta$ here is simple to consider the restriction that we mentioned earlier and to make sure the difference is less than $\epsilon$.
The reason why we could choose a bound of $\mid x - 2\mid$ is that the limit is a local feature. We could only consider the case where $x$ is near $2$. How we choose $\delta$ here is simple to consider the restriction that we mentioned earlier and to make sure the difference is less than $\epsilon$.