Chapter 2.7, Problem 56E

Proving that $\underset{x\to a}{\lim }f(x)\ne L$ Use the following definition for the nonexistence of a limit. Assume f is defined for all values of x near a, except possibly at a. We write $\underset{x\to a}{\lim }f(x)\ne L$ if for some ε > 0, there is no value of δ > 0 satisfying the condition

$|f(x)-L|<\epsilon \text{whenever}0<|x-a|<\delta .$

56. For the following function, note that $\underset{x\to 2}{\lim }f(x)\ne 3$. Find all values of ε > 0 for which the preceding condition for nonexistence is satisfied.