If we know the following about the function f: $|x-1|<0.5 => |f(x)-3|<2$
What can we say about the limit, L, if the limit of f(x) as x approaches 1 exists?
Is the only thing that we can conclude is that L = 3? Or can L be between 1 and 5, if it is the latter could you explain why that is the case?
Additionally, does the fact that the statment is $|x-1|<0.5|$ and not $0<|x-1|<0.5$ have any significance?
We can say that the lmit $1< \mathrm L <5$, but we can not conclude to a specific value of that limit Since we don't know the behaviour of function in $0.5 < x < 1.5$. We can always find a function given a number $c \in (1,5) $ such that $$\lim_{x \to 1} f(x) = c$$
For example, the function $f(x) = c$ does that work.