For the limit
lim
x → 2
(x3 − 5x + 5) = 3
illustrate the definition by finding the largest possible values of δ that correspond to ε = 0.2 and ε = 0.1.

means there exists some such that whenever , it's guaranteed that for all .

You have



Completing the square for the quadratic term yields



and by the triangle inequality,



Suppose we fix . Then if , it follows that , since



This means




So to guarantee that is within of the limit, we can choose the smaller of the two choices for , or .

Now, when , you would have , and when , you would have .

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